From 5847589e7a1178d54ef419a6c1d2a9ce82b18ad6 Mon Sep 17 00:00:00 2001 From: Tim Daly Date: Mon, 19 Sep 2016 18:52:37 -0400 Subject: [PATCH] books/bookvol10.5 add Sven Hammarling chapter Goal: Axiom Literate Programming Note that adding the Hammarling equations forced the addition of mathtools.sty... which forced a new version of \matrix... which forced a rewrite of several books. Code rot strikes. Sigh. \index{Acton, F.S.} \begin{chunk}{axiom.bib} @book{Acto70, author = "Acton, F.S.", title = "Numerical Methods that (Usually) Work", year = "1970", publisher = "Harper and Row", address = "New York, USA" } \end{chunk} \index{Acton, F.S.} \begin{chunk}{axiom.bib} @book{Acto96, author = "Acton, F.S.", title = "Real Computing Made Real: Preventing Errors in Scientific and Engineering Calculations", year = "1996", publisher = "Princeton University Press", address = "Princeton, N.J. USA", isbn = "0-691-03663-2" } \end{chunk} \index{Alefeld, G.} \index{Mayer, G.} \begin{chunk}{axiom.bib} @article{Alef00, author = "Alefeld, G. and Mayer, G.", title = "Interval analysis: Theory and applications", journal = "J. Comput. Appl. Math.", volume = "121", pages = "421-464", year = "2000" } \end{chunk} \index{Anderson, E.} \index{Bai, Z.} \index{Bischof, S.} \index{Blackford, S.} \index{Demmel, J.} \index{Dongarra, J. J.} \index{DuCroz, J.} \index{Greenbaum, A.} \index{Hammarling, S.} \index{McKenney, A.} \index{Sorensen, D. C.} \begin{chunk}{axiom.bib} @book{Ande99, author = "Anderson, E. and Bai, Z. and Bischof, S. and Blackford, S. and Demmel, J. and Dongarra, J. J. and DuCroz, J. and Greenbaum, A. and Hammarling, S. and McKenney, A. Sorensen, D. C.", title = "LAPACK Users' Guide", publisher = "SIAM", year = "1999", isbn = "0-89871-447-8", url = "www.netlib.org/lapack/lug/" } \end{chunk} \index{Bindel, D.} \index{Demmel, J.} \index{Kahan, W.} \index{Marques, O.} \begin{chunk}{axiom.bib} @article{Bind02, author = "Bindel, D. and Demmel, J. and Kahan, W. and Marques, O.", title = On computing Givens rotations reliably and efficiently", journal = "ACM Trans. Math. Software", volume = "28", pages = "206-238", year = "2002" } \end{chunk} \index{Blackford, L. S.} \index{Cleary, A.} \index{Demmel, J.} \index{Dhillon, I.} \index{Dongarra, J. J.} \index{Hammarling, S.} \index{Petitet, A.} \index{Ren, H.} \index{Stanley, K.} \index{Whaley, R. C.} \begin{chunk}{axiom.bib} @article{Blac97, author = "Blackford, L. S. and Cleary, A. and Demmel, J. and Dhillon, I. and Dongarra, J. J. and Hammarling, S. and Petitet, A. and Ren, H. and Stanley, K. and Whaley, R. C.", title = "Practical experience in the numerical dangers of heterogeneous computing", journal = "ACM Trans. Math. Software", volume = "23", pages = "133-147", year = "1997" } \end{chunk} \index{Brankin, R. W.} \index{Gladwell, I.} \begin{chunk}{axiom.bib} @article{Bran97, author = "Brankin, R. W. and Gladwell, I.", title = "rksuite\_90: Fortran 90 software for ordinary differential equation initial-value problems", journal = "ACM Trans. Math. Software", volume = "23", pages = "402-415", year = "1997" } \end{chunk} \index{Brankin, R. W.} \index{Gladwell, I.} \index{Shampine, L. F.} \begin{chunk}{axiom.bib} @techreport{Bran92, author = "Brankin, R. W. and Gladwell, I. and Shampine, L. F.", title = "RKSUITE: A suite of runge-kutta codes for the initial value problem for ODEs", year = "1992", institution = "Southern Methodist University, Dept of Math.", number = "Softreport 92-S1", type = "Technical Report" } \end{chunk} \index{Britton, J. L.} \begin{chunk}{axiom.bib} @book{Brit92, author = "Britton, J. L.", title = "Collected Works of A. M. Turing: Pure Mathematics", publisher = "North-Holland", years = "1992", isbn = "0-444-88059-3" } \end{chunk} \index{Chaitin-Chatelin, F.} \index{Fraysse, V.} \begin{chunk}{axiom.bib} @book{Chai96, author = "Chaitin-Chatelin, F. and Fraysse, V.", title = "Lectures on Finite Precision Computations", publisher = "SIAM", year = "1996", isbn = "0-89871-358-7" } \end{chunk} \index{Chan, T. F.} \index{Golub, G. H.} \index{LeVeque, R. J.} \begin{chunk}{axiom.bib} @article{Chan83, author = "Chan, T. F. and Golub, G. H. and LeVeque, R. J.", title = "Algorithms for computing the sample variance: Analysis and recommendations", journal = "The American Statistician", volume = "37", pages = "242-247", year = "1983" } \end{chunk} \index{Cools, R.} \index{Haegemans, A.} \begin{chunk}{axiom.bib} @article{Cool03, author = "Cools, R. and Haegemans, A.", title = "Algorithm 824: CUBPACK: A package for automatic cubature; framework description", journal = "ACM Trans. Math. Software", volume = "29", pages = "287-296", year = "2003" } \end{chunk} \index{Cox, M. G.} \index{Dainton, M. P.} \index{Harris, P. M.} \begin{chunk}{axiom.bib} @techreport{Coxx00, author = "Cox, M. G. and Dainton, M. P. and Harris, P. M.", title = "Testing spreadsheets and other packages used in metrology: Testing functions for the calculation of standard deviation", year = "2000", institution = "National Physical Lab, Teddington, Middlesex UK", type = "Technical Report", number = "NPL Report CMSC07/00" } \end{chunk} \index{Dodson, D. S.} \begin{chunk}{axiom.bib} @article{Dods83, author = "Dodson, D. S.", title = "Corrigendum: Remark on 'Algorithm 539: Basic Linear Algebra Subroutines for FORTRAN usage", journal = "ACM Trans. Math. Software", volume = "9", pages = "140", year = "1983" } \end{chunk} \index{Dodson, D. S.} \index{Grimes, R. G.} \begin{chunk}{axiom.bib} @article{Dods82, author = "Dodson, D. S. and Grimes, R. G.", title = "Remark on algorithm 539: Basic Linear Algebra Subprograms for Fortran usage", journal = "ACM Trans. Math. Software", volume = "8", pages = "403-404", year = "1982" } \end{chunk} \index{Dongarra, J. J.} \index{DuCroz, J.} \index{Hammarling, S.} \index{Hanson, R. J.} \begin{chunk}{axiom.bib} @article{Dong88, author = "Dongarra, J. J. and DuCroz, J. and Hammarling, S. and Hanson, R. J.", title = "An extended set of FORTRAN Basic Linear Algebra Subprograms", journal = "ACM Trans. Math. Software", volume = "14", pages = "1-32", year = "1988" } \end{chunk} \index{Dongarra, J.} \index{DuCroz, J.} \index{Duff, I. S.} \index{Hammarling, S.} \begin{chunk}{axiom.bib} @article{Dong90, author = "Dongarra, J. and DuCroz, J. and Duff, I. S. and Hammarling, S.", title = "A set of Level 3 Basic Linear Algebra Subprograms", journal = "ACM Trans. Math. Software", volume = "16", pages = "1-28", year = "1990" } \end{chunk} \index{Dubrulle, A. A.} \begin{chunk}{axiom.bib} @article{Dubr83, author = "Dubrulle, A. A.", title = "A class of numerical methods for the computation of Pythagorean sums", journal = "IBM J. Res. Develop.", volume = "27", number = "6", pages = "582-589", year = "1983" } \end{chunk} \index{Einarsson, B.} \begin{chunk}{axiom.bib} @book{Eina05, author = "Einarsson, B.", title = "Accuracy and Reliability in Scientific Computing", publisher = "SIAM", year = "2005", isbn = "0-89871-584-9", url = "http://www.nsc.liu.se/wg25/book/" } \end{chunk} \index{Forsythe, G. E.} \begin{chunk}{axiom.bib} @article{Fors70, author = "Forsythe, G. E.", title = "Pitfalls in computations, or why a math book isn't enough", journal = "Amer. Math. Monthly", volume = "9", pages = "931-995", year = "1970" } \end{chunk} \index{Forsythe, G. E.} \begin{chunk}{axiom.bib} @incollection{Fors69, author = "Forsythe, G. E.", title = "What is a satisfactory quadratic equation solver", booktitle = "Constructive Aspects of the Fundamental Theorem of Algebra", pages = "53-61", publisher = "Wiley", year = "1969" } \end{chunk} \index{Fox, L.} \begin{chunk}{axiom.bib} @article{Foxx71, author = "Fox, L.", title = "How to get meaningless answers in scientific computations (and what to do about it)", journal = "IMA Bulletin", volume = "7", pages = "296-302", year = "1971" } \end{chunk} \index{Givens, W.} \begin{chunk}{axiom.bib} @techreport{Give54, author = "Givens, W.", title = "Numerical computation of the characteristic values of a real symmetric matrix", year = "1954", institution = "Oak Ridge National Laboratory", type = "Technical Report", number = "ORNL-1574" } \end{chunk} \index{Golub, G.H.} \begin{chunk}{axiom.bib} @article{Golu65, author = "Golub, G.H.", title = "Numerical methods for solving linear least squares problems", journal = "Numer. Math.", volume = "7", pages = "206-216", year = "1965" } \end{chunk} \index{Golub, Gene H.} \index{Van Loan, Charles F.} \begin{chunk}{axiom.bib} @book{Golu89, author = "Golub, Gene H. and Van Loan, Charles F.", title = "Matrix Computations", publisher = "Johns Hopkins University Press", year = "1989", isbn = "0-8018-3772-3" } \end{chunk} \index{Golub, Gene H.} \index{Van Loan, Charles F.} \begin{chunk}{axiom.bib} @book{Golu96, author = "Golub, Gene H. and Van Loan, Charles F.", title = "Matrix Computations", publisher = "Johns Hopkins University Press", isbn = "978-0-8018-5414-9", year = "1996" } \end{chunk} \index{Hammarling S.} \begin{chunk}{axiom.bib} @article{Hamm85, author = "Hammarling S.", title = " The Singular Value Decomposition in Multivariate Statistics", journal = "ACM Signum Newsletter", volume = "20", number = "3", pages = "2--25", year = "1985" } \end{chunk} \index{Hammarling, Sven} \begin{chunk}{axiom.bib} @book{Hamm05, author = "Hammarling, Sven", title = "An Introduction to the Quality of Computed Solutions", booktitle = "Accuracy and Reliability in Scientific Computing", year = "2005", publisher = "SIAM", pages = "43-76", url = "http://eprints.ma.man.ac.uk/101/", paper = "Hamm05.pdf" } \end{chunk} \index{Hargreaves, G.} \begin{chunk}{axiom.bib} @mastersthesis{Harg02, author = "Hargreaves, G.", title = "Interval analysis in MATLAB", school = "University of Manchester, Dept. of Mathematics", year = "2002" } \end{chunk} \index{Higham, Nicholas J.} \begin{chunk}{axiom.bib} @book{High02, author = "Higham, Nicholas J.", title = "Accuracy and stability of numerical algorithms", publisher = "SIAM", isbn = "0-89871-521-0", year = "2002" } \end{chunk} \index{Higham, Nicholas J.} \begin{chunk}{axiom.bib} @article{High88, author = "Higham, Nicholas J.", title = "FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation", journal = "ACM Trans. Math. Soft", volume = "14", number = "4", pages = "381-396", year = "1988" } \end{chunk} \begin{chunk}{axiom.bib} @misc{IEEE85, author = "IEEE", title = "ANSI/IEEE Standard for Binary Floating Point Arithmetic: Std 754-1985", publisher = "IEEE Press", year = "1985" } \end{chunk} \begin{chunk}{axiom.bib} @misc{IEEE87, author = "IEEE", title = "ANSI/IEEE Standard for Radix Independent Floating Point Arithmetic: Std 854-1987", publisher = "IEEE Press", year = "1987" } \end{chunk} \index{Kn\"usel, L.} \begin{chunk}{axiom.bib} @article{Knus98, author = {Kn\"usel, L.}, title = "On the accuracy of statistical distributions in Microsoft Excel 97", journal = "Comput. Statist. Data Anal.", volume = "26", pages = "375-377", year = "1998" } \end{chunk} \index{Kreinovich, V.} \begin{chunk}{axiom.bib} @misc{Krei05, author = "Kreinovich, V.", title = "Interval cmoputations", year = "2005", url = "http://www.cs.utep.edu/interval-comp/" } \end{chunk} \index{Lawson, C. L.} \index{Hanson, R. J.} \begin{chunk}{axiom.bib} @book{Laws75, author = "Lawson, C. L. and Hanson, R. J.", title = "Solving Least Squares Problems", publisher = "Prentice-Hall", year = "1974" } \end{chunk} \index{Lawson, C. L.} \index{Hanson, R. J.} \begin{chunk}{axiom.bib} @book{Laws95, author = "Lawson, C. L. and Hanson, R. J.", title = "Solving Least Squares Problems", publisher = "SIAM", isbn = "0-89871-356-0", year = "1995" } \end{chunk} \index{Lawson, C. L.} \index{Hanson, R. J.} \index{Kincaid, D.} \index{Krogh, F. T.} \begin{chunk}{axiom.bib} @article{Laws79, author = "Lawson, C. L. and Hanson, R. J. and Kincaid, D. and Krogh, F. T.", title = "Basic Linear Algebra Subprograms for FORTRAN usage", journal = "ACM Trans. Math. Software", volume = "5", pages = "308-323", year = "1979" } \end{chunk} \index{Martin, R. S.} \index{Wilkinson, J. H.} \begin{chunk}{axiom.bib} @article{Mart68, author = "Martin, R. S. and Wilkinson, J. H.", title = "Similarity reduction ofa general matrix to Hessenberg form", journal = "Numer. Math.", volume = "12", pages = "349-368", year = "1968" } \end{chunk} \begin{chunk}{axiom.bib} @misc{Math05, author = "MathWorks", title = "MATLAB", publisher = "The Mathworks, Inc.", url = "http://www.mathworks.com" } \end{chunk} \index{McCullough, B. D.} \index{Wilson, B.} \begin{chunk}{axiom.bib} @article{Mccu02, author = "McCullough, B. D. and Wilson, B.", title = "On the accuracy of statistical procedures in Microsoft Excel 2000 and Excel XP", journal = "Comput. Statist. Data Anal.", volume = "40", pages = "713-721", year = "2002" } \end{chunk} \index{McCullough, B. D.} \index{Wilson, B.} \begin{chunk}{axiom.bib} @article{Mccu99, author = "McCullough, B. D. and Wilson, B.", title = "On the accuracy of statistical procedures in Microsoft Excel 97", journal = "Comput. Statist. Data Anal.", volume = "31", pages = "27-37", year = "1999" } \end{chunk} \index{Metcalf, M.} \index{Reid, J. K.} \begin{chunk}{axiom.bib} @book{Metc96, author = "Metcalf, M. and Reid, J. K.", title = "Fortran 90/95 Explained", publisher = "Oxford University Press", year = "1996" } \end{chunk} \index{Metcalf, M.} \index{Reid, J. K.} \index{Cohen, M.} \begin{chunk}{axiom.bib} @book{Metc04, author = "Metcalf, M. and Reid, J. K. and Cohen, M.", title = "Fortran 95/2003 Explained", publisher = "Oxford University Press", year = "2004", isbn = "0-19-852693-8" } \end{chunk} \index{Moler, C.} \index{Morrison, D.} \begin{chunk}{axiom.bib} @article{Mole83, author = "Moler, C. and Morrison, D.", title = "Replacing square roots by Pythagorena sums", journal = "IBM J. Res. Develop.", volume = "27", number = "6", pages = "577-581", year = "1983" } \end{chunk} \index{Moore, R. E.} \begin{chunk}{axiom.bib} @books{Moor79, author = "Moore, R. E.", title = "methods and Applications of Interval Analysis", publisher = "SIAM", year = "1979" } \end{chunk} \begin{chunk}{axiom.bib} @misc{NAGa05, author = "Numerical Algorithms Group", title = "The NAG Library", url = "http://www.nag.co.uk/numeric", year = "2005" } \end{chunk} \begin{chunk}{axiom.bib} @misc{NAGb05, author = "Numerical Algorithms Group", title = "The NAG Fortran Library Manual", url = "http://www.nag.co.uk/numeric/fl/manual/html/FLlibrarymanual.asp", year = "2005" } \end{chunk} \index{Overton, M. L.} \begin{chunk}{axiom.bib} @book{Over01, author = "Overton, M. L.", title = "Numerical Computing with IEEE Floating Point Arithmetic", publisher = "SIAM", year = "2001", isbn = "0-89871-482-6" } \end{chunk} \index{Piessens, R.} \index{de Doncker-Kapenga, E.}, \index{\"Uberhuber, C. W.} \index{Kahaner, D. K.} \begin{chunk}{axiom.bib} @book{Pies83, author = {Piessens, R. and de Doncker-Kapenga, E. and \"Uberhuber, C. W. and Kahaner, D. K.}, title = "QUADPACK - A Subroutine Package for Automatic Integration", publisher = "Springer-Verlag", year = "1983" } \end{chunk} \index{Priest, D. M.} \begin{chunk}{axiom.bib} @article{Prie04, author = "Priest, D. M.", title = "Efficient scaling for complex division", journal = "ACM Trans. Math. Software", volume = "30", pages = "389-401", year = "2004" } \end{chunk} \index{Rump, S. M.} \begin{chunk}{axiom.bib} @InProceedings{Rump99, author = "Rump, S. M.", title = "INTLAB - INTerval LABoratory", booktitle = "Developments in Reliable Computing", pages = "77-104", publisher = "Kluwer Academic", year = "1999" } \end{chunk} \index{Shampine, L. F.} \index{Gladwell, I.} \begin{chunk}{axiom.bib} @InProceedings{Sham92, author = "Shampine, L. F. and Gladwell, I.", title = "The next generation of runge-kutta codes", booktitle = "Computational Ordinary Differential Equations", pages = "145-164", publisher = "Oxford University Press", year = "1992" } \end{chunk} \index{Smith, R. L.} \begin{chunk}{axiom.bib} @article{Smit62, author = "Smith, R. L.", title = "Algorithm 116: Complex division", journal = "Communs. Ass. comput. Mach.", volume = "5", pages = "435", year = "1962" } \end{chunk} \index{Stewart, G. W.} \begin{chunk}{axiom.bib} @book{Stew98, author = "Stewart, G. W.", title = "Matrix Algorithms: Basic Decompositions, volume I", publisher = "SIAM", year = "1998", isbn = "0-89871-414-1" } \end{chunk} \index{Stewart, G. W.} \begin{chunk}{axiom.bib} @article{Stew85, author = "Stewart, G. W.", title = "A note on complex division", journal = "ACM Trans. Math. Software", volume = "11", pages = "238-241", year = "1985" } \end{chunk} \index{Stewart, G. W.} \index{Sun, J.} \begin{chunk}{axiom.bib} @book{Stew90, author = "Stewart, G. W. and Sun, J.", title = "Matrix Perturbation Theory", publisher = "Academic Press", year = "1990" } \end{chunk} \index{Turing, A. M.} \begin{chunk}{axiom.bib} @article{Turi48, author = "Turing, A. M.", title = "Rounding-off errors in matrix processes", journal = "Q. J. Mech. Appl. Math.", volume = "1", pages = "287-308", year = "1948" } \end{chunk} \index{Vignes, J.} \begin{chunk}{axiom.bib} @article{Vign93, author = "Vignes, J.", title = "A stochastic arithmetic for reliable scientific computation", jouirnal = "Math. and Comp. in Sim.", volume = "25", pages = "233-261", year = "1993" } \end{chunk} \index{Wilkinson, J. H.} \begin{chunk}{axiom.bib} @book{Wilk63, author = "Wilkinson, J. H.", title = "Rounding Erroors in Algebraic Processes", publisher = "HMSO", series = "Notes on Applied Science, No. 32", year = "1963" } \end{chunk} \index{Wilkinson, J. H.} \begin{chunk}{axiom.bib} @book{Wilk65, author = "Wilkinson, J. H.", title = "The Algebraic Eigenvalue Problem", publisher = "Oxford University Press", year = "1965" } \end{chunk} \index{Wilkinson, J. H.} \begin{chunk}{axiom.bib} @InProceedings{Wilk84, author = "Wilkinson, J. H.", title = "The perfidious polynomial", booktitle = "Studies in Numerical Analysis", volume = "24", chapter = "1", pages = "1-28", year = "1984" } \end{chunk} \index{Wilkinson, J. H.} \begin{chunk}{axiom.bib} @article{Wilk86, author = "Wilkinson, J. H.", title = "Error analysis revisited", journal = "IMA Bulletin", volume = "22", pages = "192-200", year = "1986" } \end{chunk} \index{Wilkinson, J. H.} \begin{chunk}{axiom.bib} @article{Wilk61, author = "Wilkinson, J. H.", title = "Error analysis of diret methods of matrix inversion", journal = "J. ACM", volume = "8", pages = "281-330", year = "1961" } \end{chunk} \index{Wilkinson, J. H.} \begin{chunk}{axiom.bib} @article{Wilk85, author = "Wilkinson, J. H.", title = "The state of the art in error analysis", journal = "NAG Newsletter", volume = "2/85", pages = "5-28", year = "1985" } \end{chunk} \index{Wilkinson, J. H.} \begin{chunk}{axiom.bib} @article{Wilk60, author = "Wilkinson, J. H.", title = "Error analysis of floating-point computation", journal = "Numer. Math.", volume = "2", pages = "319-340", year = "1960" } \end{chunk} \index{Wilkinson, J. H.} \index{Reinsch, C.} \begin{chunk}{axiom.bib} @book{Wilk71, author = "Wilkinson, J. H.", title = "Handbook for Automatic Computation, V2, Linear Algebra", publisher = "Springer-Verlag", year = "1971" } \end{chunk} --- books/Makefile.pamphlet | 8 +- books/bookheader.tex | 1 + books/bookvol0.pamphlet | 5 +- books/bookvol1.pamphlet | 5 +- books/bookvol10.3.pamphlet | 1014 ++++++++++++---------- books/bookvol10.5.pamphlet | 1821 ++++++++++++++++++++++++++++++++++++++++ books/bookvol10.pamphlet | 144 ++++- books/bookvol5.pamphlet | 197 +---- books/bookvolbib.pamphlet | 1501 ++++++++++++++++++++++++++------- books/bookvolbug.pamphlet | 605 +++++++++----- books/mathtools.sty | 1650 ++++++++++++++++++++++++++++++++++++ books/mhsetup.sty | 175 ++++ books/ps/v105hammfig1.eps | 627 ++++++++++++++ books/ps/v105hammfig2.eps | 452 ++++++++++ books/ps/v105hammfig3.eps | 1003 ++++++++++++++++++++++ books/ps/v105hammfig4.eps | 1014 ++++++++++++++++++++++ books/ps/v105hammfig5.eps | 1031 +++++++++++++++++++++++ books/ps/v105hammfig6.eps | 981 ++++++++++++++++++++++ books/ps/v105hammfig7.eps | 1055 +++++++++++++++++++++++ books/ps/v105hammtable2.eps | 1366 ++++++++++++++++++++++++++++++ changelog | 20 + patch | 907 ++++++++++++++++++++- src/axiom-website/patches.html | 2 + 23 files changed, 14449 insertions(+), 1135 deletions(-) create mode 100644 books/mathtools.sty create mode 100644 books/mhsetup.sty create mode 100644 books/ps/v105hammfig1.eps create mode 100644 books/ps/v105hammfig2.eps create mode 100644 books/ps/v105hammfig3.eps create mode 100644 books/ps/v105hammfig4.eps create mode 100644 books/ps/v105hammfig5.eps create mode 100644 books/ps/v105hammfig6.eps create mode 100644 books/ps/v105hammfig7.eps create mode 100644 books/ps/v105hammtable2.eps diff --git a/books/Makefile.pamphlet b/books/Makefile.pamphlet index 7264a15..a84dce6 100644 --- a/books/Makefile.pamphlet +++ b/books/Makefile.pamphlet @@ -156,9 +156,7 @@ ${COOKBOOK}/%.pdf: ${COOKIN}/%.pamphlet @ echo making ${COOKBOOK}/$*.pdf from ${COOKIN}/$*.pamphlet @ echo =========================================== @(cd ${COOKBOOK} ; \ - cp ${BOOKS}/axiom.sty ${COOKBOOK} ; \ - cp ${BOOKS}/bbold.sty ${COOKBOOK} ; \ - cp ${BOOKS}/appendix.sty ${COOKBOOK} ; \ + cp ${BOOKS}/*.sty ${COOKBOOK} ; \ cp ${COOKIN}/$*.pamphlet ${COOKBOOK} ; \ ${RM} $*.toc ; \ if [ -z "${NOISE}" ] ; then \ @@ -185,9 +183,7 @@ ${PDF}/%.pdf: ${IN}/%.pamphlet ${PDF}/axiom.bib ${PDF}/axiom.bst @ echo making ${PDF}/$*.pdf from ${IN}/$*.pamphlet @ echo =========================================== @(cd ${PDF} ; \ - cp ${BOOKS}/axiom.sty ${PDF} ; \ - cp ${BOOKS}/bbold.sty ${PDF} ; \ - cp ${BOOKS}/appendix.sty ${PDF} ; \ + cp ${BOOKS}/*.sty ${PDF} ; \ cp ${IN}/$*.pamphlet ${PDF} ; \ cp ${IN}/bookheader.tex ${PDF} ; \ cp -pr ${IN}/ps ${PDF} ; \ diff --git a/books/bookheader.tex b/books/bookheader.tex index 12c339d..453b0a4 100644 --- a/books/bookheader.tex +++ b/books/bookheader.tex @@ -2,6 +2,7 @@ \hypersetup{colorlinks=true,linkcolor=blue,pdfborderstyle={/S/U/W 1}, citecolor=red} \usepackage[toc,page]{appendix} +\usepackage{mathtools} \usepackage{amssymb} \usepackage{axiom} \usepackage{makeidx} diff --git a/books/bookvol0.pamphlet b/books/bookvol0.pamphlet index 324af8a..cc11b7f 100644 --- a/books/bookvol0.pamphlet +++ b/books/bookvol0.pamphlet @@ -1583,7 +1583,10 @@ and power series, to name a few. The Axiom interpreter reads user input then builds whatever types it needs to perform the indicated computations. For example, to create the matrix -$$M = \pmatrix{x^2+1&0\cr0&x / 2\cr}$$ +\[M=\left(\begin{array}{cc} +x^2+1 & 0\\ +0 & x/2 +\end{array}\right)\] using the command: \spadcommand{M = [ [x**2+1,0],[0,x / 2] ]::Matrix(POLY(FRAC(INT)))} diff --git a/books/bookvol1.pamphlet b/books/bookvol1.pamphlet index 369c871..cd38073 100644 --- a/books/bookvol1.pamphlet +++ b/books/bookvol1.pamphlet @@ -816,7 +816,10 @@ and power series, to name a few. The Axiom interpreter reads user input then builds whatever types it needs to perform the indicated computations. For example, to create the matrix -$$M = \pmatrix{x^2+1&0\cr0&x / 2\cr}$$ +\[M=\left(\begin{array}{cc} +x^2+1 & 0\\ +0 & x / 2 +\end{array}\right)\] using the command: \spadcommand{M = [ [x**2+1,0],[0,x / 2] ]::Matrix(POLY(FRAC(INT)))} diff --git a/books/bookvol10.3.pamphlet b/books/bookvol10.3.pamphlet index c73c31c..1cf4911 100644 --- a/books/bookvol10.3.pamphlet +++ b/books/bookvol10.3.pamphlet @@ -35121,11 +35121,12 @@ where {\bf i}, {\bf j}, and {\bf k} are unit vectors along the $x$, $y$, and $z$ coordinate axes, respectively, is represented in homogeneous coordinates as a column matrix -$${\bf v} = \left[\matrix{{\bf x}\cr - {\bf y}\cr - {\bf z}\cr - {\bf w}\cr} - \right]\eqno(1.2)$$ +\[{\bf v} = \left[\begin{array}{c} +{\bf x}\\ +{\bf y}\\ +{\bf z}\\ +{\bf w} +\end{array}\right]\eqno(1.2)\] \noindent where @@ -35171,9 +35172,11 @@ This definition is easily remembered as the expansion of the determinant $${\bf a} \times {\bf b} = - \left|\matrix{{\bf i}&{\bf j}&{\bf k}\cr - {a_x}&{a_y}&{a_z}\cr - {b_x}&{b_y}&{b_z}\cr}\right|\eqno(1.7)$$ + \left|\begin{array}{ccc} +{\bf i} & {\bf j} & {\bf k}\\ +{a_x} & {a_y} & {a_z}\\ +{b_x} & {b_y} & {b_z} +\end{array}\right|\eqno(1.7)$$ \subsection{Planes} A plane is represented as a row matrix @@ -35221,42 +35224,46 @@ $${\rm {\ \ \ \ \ or\ as\ \ \ }} \bigp = [0,0,-100,100]\eqno(1.15)$$ \noindent A point ${\bf v} = [10,20,1,1]$ should lie in this plane -$$[0,0,-100,100]\left[\matrix{10\cr - 20\cr - 1\cr - 1\cr} - \right] +$$[0,0,-100,100]\left[\begin{array}{c} +10\\ +20\\ +1 +1 +\end{array}\right] = 0\eqno(1.16)$$ \noindent or -$$[0,0,1,-1]\left[\matrix{ -5\cr - -10\cr - -.5\cr - -.5\cr} - \right] +$$[0,0,1,-1]\left[\begin{array}{c} +-5\\ +-10\\ +-.5\\ +-.5 +\end{array}\right] = 0\eqno(1.17)$$ \noindent The point ${\bf v} = [0,0,2,1]$ lies above the plane -$$[0,0,2,-2]\left[\matrix{0\cr - 0\cr - 2\cr - 1\cr} - \right] +$$[0,0,2,-2]\left[\begin{array}{c} +0\\ +0\\ +2\\ +1 +\end{array}\right] = 2\eqno(1.18)$$ and $\bigp{\bf v}$ is indeed positive, indicating that the point is outside the plane in the direction of the outward pointing normal. A point ${\bf v} = [0,0,0,1]$ lies below the plane -$$[0,0,1,-1]\left[\matrix{0\cr - 0\cr - 0\cr - 1\cr} - \right] +$$[0,0,1,-1]\left[\begin{array}{c} +0\\ +0\\ +0\\ +1 +\end{array}\right] = -1\eqno(1.19)$$ \noindent @@ -35297,38 +35304,43 @@ The transformation {\bf H} corresponding to a translation by a vector $a{\bf i} + b{\bf j} + c{\bf k}$ is $${\bf H} = {\bf Trans(a,b,c)} = - \left[\matrix{1&0&0&a\cr - 0&1&0&b\cr - 0&0&1&c\cr - 0&0&0&1\cr} - \right]\eqno(1.24)$$ + \left[\begin{array}{cccc} +1 & 0 & 0 & a\\ +0 & 1 & 0 & b\\ +0 & 0 & 1 & c\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.24)$$ \noindent Given a vector ${\bf u} = [x,y,z,w]^{\rm T}$ the transformed vector {\bf v} is given by $${\bf H} = {\bf Trans(a,b,c)} = - \left[\matrix{1&0&0&a\cr - 0&1&0&b\cr - 0&0&1&c\cr - 0&0&0&1\cr} - \right] - \left[\matrix{x\cr - y\cr - z\cr - w\cr} - \right]\eqno(1.25)$$ - -$${\bf v} = \left[\matrix{x + aw\cr - y + bw\cr - z + cw\cr - w\cr} - \right] - = \left[\matrix{x/w + a\cr - y/w + b\cr - z/w + c\cr - 1\cr} - \right]\eqno(1.26)$$ + \left[\begin{array}{cccc} +1 & 0 & 0 & a\\ +0 & 1 & 0 & b\\ +0 & 0 & 1 & c\\ +0 & 0 & 0 & 1 +\end{array} \right] + \left[\begin{array}{c} +x\\ +y\\ +z\\ +w +\end{array}\right]\eqno(1.25)$$ + +$${\bf v} = \left[\begin{array}{c} +x + aw\\ +y + bw\\ +z + cw\\ +w +\end{array}\right] + = \left[\begin{array}{c} +x/w + a\\ +y/w + b\\ +z/w + c\\ +1 +\end{array}\right]\eqno(1.26)$$ \noindent The translation may also be interpreted as the addition of the two @@ -35341,78 +35353,90 @@ manner as points and planes. Consider the vector $2{\bf i} + 3{\bf j} + 2{\bf k}$ translated by, or added to\\ 4{\bf i} - 3{\bf j} + 7{\bf k} -$$\left[\matrix{6\cr - 0\cr - 9\cr - 1\cr} - \right] = - \left[\matrix{1 & 0 & 0 & 4\cr - 0 & 1 & 0 & -3\cr - 0 & 0 & 1 & 7\cr - 0 & 0 & 0 & 1\cr} - \right] - \left[\matrix{2\cr - 3\cr - 2\cr - 1\cr} - \right]\eqno(1.27)$$ +$$\left[\begin{array}{c} +6\\ +0\\ +9\\ +1 +\end{array}\right] = + \left[\begin{array}{cccc} +1 & 0 & 0 & 4\\ +0 & 1 & 0 & -3\\ +0 & 0 & 1 & 7\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{c} +2\\ +3\\ +2\\ +1 +\end{array}\right]\eqno(1.27)$$ \noindent If we multiply the transmation matrix elements by, say, -5, and the vector elements by 2, we obtain -$$\left[\matrix{-60\cr - 0\cr - -90\cr - -10\cr} - \right] = - \left[\matrix{-5 & 0 & 0 & -20\cr - 0 & -5 & 0 & 15\cr - 0 & 0 & -5 & -35\cr - 0 & 0 & 0 & -5\cr} - \right] - \left[\matrix{4\cr - 6\cr - 4\cr - 2\cr} - \right]\eqno(1.28)$$ +$$\left[\begin{array}{c} +-60\\ +0\\ +-90\\ +-10 +\end{array}\right] = + \left[\begin{array}{cccc} +-5 & 0 & 0 & -20\\ +0 & -5 & 0 & 15\\ +0 & 0 & -5 & -35\\ +0 & 0 & 0 & -5 +\end{array}\right] + \left[\begin{array}{c} +4\\ +6\\ +4\\ +2 +\end{array}\right]\eqno(1.28)$$ \noindent which corresponds to the vector $[6,0,9,1]^{\rm T}$ as before. The point $[2,3,2,1]$ lies in the plane $[1,0,0,-2]$ -$$[1,0,0,-2]\left[\matrix{2\cr - 3\cr - 2\cr - 1\cr} - \right] = 0\eqno(1.29)$$ +$$[1,0,0,-2]\left[\begin{array}{c} +2\\ +3\\ +2\\ +1 +\end{array}\right] = 0\eqno(1.29)$$ \noindent The transformed point is, as we have already found, $[6,0,9,1]^{\rm T}$. We will now compute the transformed plane. The inverse of the transform is -$$\left[\matrix{1 & 0 & 0 & -4\cr - 0 & 1 & 0 & 3\cr - 0 & 0 & 1 & -7\cr - 0 & 0 & 0 & 1\cr}\right]$$ +$$\left[\begin{array}{cccc} +1 & 0 & 0 & -4\\ +0 & 1 & 0 & 3\\ +0 & 0 & 1 & -7\\ +0 & 0 & 0 & 1 +\end{array}\right]$$ \noindent and the transformed plane -$$[1\ 0\ 0\ -6] = [1\ 0\ 0\ -2]\left[\matrix{1 & 0 & 0 & -4\cr - 0 & 1 & 0 & 3\cr - 0 & 0 & 1 & -7\cr - 0 & 0 & 0 & 1\cr} - \right]\eqno(1.30)$$ +$$[1\ 0\ 0\ -6] = [1\ 0\ 0\ -2]\left[\begin{array}{cccc} +1 & 0 & 0 & -4\\ +0 & 1 & 0 & 3\\ +0 & 0 & 1 & -7\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.30)$$ \noindent Once again the transformed point lies in the transformed plane -$$[1\ 0\ 0\ -6] \left[\matrix{6\cr - 0\cr - 9\cr - 1\cr}\right] = 0\eqno(1.31)$$ +$$[1\ 0\ 0\ -6] \left[\begin{array}{c} +6\\ +0\\ +9\\ +1 +\end{array}\right] = 0\eqno(1.31)$$ The general translation operation can be represented in Axiom as @@ -35431,10 +35455,12 @@ The transformations corresponding to rotations about the $x$, $y$, and $z$ axes by an angle $\theta$ are $${\bf Rot(x,\theta)} = - \left[\matrix{1 & 0 & 0 & 0\cr - 0 & {cos\ \theta} & {-sin\ \theta} & 0\cr - 0 & {sin\ \theta} & {cos\ \theta} & 0\cr - 0 & 0 & 0 & 1}\right] + \left[\begin{array}{cccc} +1 & 0 & 0 & 0\\ +0 & {cos\ \theta} & {-sin\ \theta} & 0\\ +0 & {sin\ \theta} & {cos\ \theta} & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] \eqno(1.32)$$ Rotations can be described in Axiom as functions that return @@ -35461,10 +35487,12 @@ The Axiom code for ${\bf Rot(x,degree)}$ is \end{chunk} $${\bf Rot(y,\theta)} = - \left[\matrix{{cos\ \theta} & 0 & {sin\ \theta} & 0\cr - 0 & 1 & 0 & 0\cr - {-sin\ \theta} & 0 & {cos\ \theta} & 0\cr - 0 & 0 & 0 & 1\cr}\right] + \left[\begin{array}{cccc} +{cos\ \theta} & 0 & {sin\ \theta} & 0\\ +0 & 1 & 0 & 0\\ +{-sin\ \theta} & 0 & {cos\ \theta} & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] \eqno(1.33)$$ \noindent @@ -35483,10 +35511,12 @@ The Axiom code for ${\bf Rot(y,degree)}$ is \end{chunk} $${\bf Rot(z,\theta)} = - \left[\matrix{{cos\ \theta} & {-sin\ \theta} & 0 & 0\cr - {sin\ \theta} & {cos\ \theta} & 0 & 0\cr - 0 & 0 & 1 & 0\cr - 0 & 0 & 0 & 1}\right] + \left[\begin{array}{cccc} +{cos\ \theta} & {-sin\ \theta} & 0 & 0\\ +{sin\ \theta} & {cos\ \theta} & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] \eqno(1.34)$$ \noindent @@ -35510,41 +35540,47 @@ rotating it $90^\circ$ about the ${\bf z}$ axis to ${\bf v}$? The transform is obtained from Equation 1.34 with $sin\ \theta = 1$ and $cos\ \theta = 0$. -$$\left[\matrix{-3\cr - 7\cr - 2\cr - 1\cr} - \right] = - \left[\matrix{0 & -1 & 0 & 0\cr - 1 & 0 & 0 & 0\cr - 0 & 0 & 1 & 0\cr - 0 & 0 & 0 & 1\cr} - \right] - \left[\matrix{7\cr - 3\cr - 2\cr - 1\cr} - \right]\eqno(1.35)$$ +$$\left[\begin{array}{c} +-3\\ +7\\ +2\\ +1 +\end{array}\right] = + \left[\begin{array}{cccc} +0 & -1 & 0 & 0\\ +1 & 0 & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{c} +7\\ +3\\ +2\\ +1 +\end{array}\right]\eqno(1.35)$$ \noindent Let us now rotate {\bf v} $90^\circ$ about the $y$ axis to {\bf w}. The transform is obtained from Equation 1.33 and we have -$$\left[\matrix{2\cr - 7\cr - 3\cr - 1\cr} - \right] = - \left[\matrix{ 0 & 0 & 1 & 0\cr - 0 & 1 & 0 & 0\cr - -1 & 0 & 0 & 0\cr - 0 & 0 & 0 & 1\cr} - \right] - \left[\matrix{-3\cr - 7\cr - 2\cr - 1\cr} - \right]\eqno(1.36)$$ +$$\left[\begin{array}{c} +2\\ +7\\ +3\\ +1 +\end{array}\right] = + \left[\begin{array}{cccc} +0 & 0 & 1 & 0\\ +0 & 1 & 0 & 0\\ +-1 & 0 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{c} +-3\\ +7\\ +2\\ +1 +\end{array}\right]\eqno(1.36)$$ \noindent If we combine these two rotations we have @@ -35560,39 +35596,48 @@ obtain $${\bf w} = {\bf Rot(y,90)}\ {\bf Rot(z,90)}\ {\bf u}\eqno(1.39)$$ $${\bf Rot(y,90)}\ {\bf Rot(z,90)} = - \left[\matrix{ 0 & 0 & 1 & 0\cr - 0 & 1 & 0 & 0\cr - -1 & 0 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right] - \left[\matrix{0 & -1 & 0 & 0\cr - 1 & 0 & 0 & 0\cr - 0 & 0 & 1 & 0\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.40)$$ + \left[\begin{array}{cccc} +0 & 0 & 1 & 0\\ +0 & 1 & 0 & 0\\ +-1 & 0 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{cccc} +0 & -1 & 0 & 0\\ +1 & 0 & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.40)$$ $${\bf Rot(y,90)}\ {\bf Rot(z,90)} = - \left[\matrix{0 & 0 & 1 & 0\cr - 1 & 0 & 0 & 0\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.41)$$ + \left[\begin{array}{cccc} +0 & 0 & 1 & 0\\ +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.41)$$ \noindent thus -$${\bf w} = \left[\matrix{2\cr - 7\cr - 3\cr - 1}\right] - = \left[\matrix{0 & 0 & 1 & 0\cr - 1 & 0 & 0 & 0\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 0 & 1}\right] - \left[\matrix{7\cr - 3\cr - 2\cr - 1}\right]\eqno(1.42)$$ +$${\bf w} = \left[\begin{array}{c} +2\\ +7\\ +3\\ +1 +\end{array}\right] + = \left[\begin{array}{cccc} +0 & 0 & 1 & 0\\ +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{c} +7\\ +3\\ +2\\ +1 +\end{array}\right]\eqno(1.42)$$ \noindent as we obtained before. @@ -35602,40 +35647,46 @@ the $y$ axis and then $90^\circ$ about the $z$ axis, we obtain a different position $${\bf Rot(z,90)}{\bf Rot(y,90)} = - \left[\matrix{0 & -1 & 0 & 0\cr - 1 & 0 & 0 & 0\cr - 0 & 0 & 1 & 0\cr - 0 & 0 & 0 & 1} - \right] - \left[\matrix{ 0 & 0 & 1 & 0\cr - 0 & 1 & 0 & 0\cr - -1 & 0 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right] - = \left[\matrix{ 0 & -1 & 0 & 0\cr - 0 & 0 & 1 & 0\cr - -1 & 0 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.43)$$ + \left[\begin{array}{cccc} +0 & -1 & 0 & 0\\ +1 & 0 & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{cccc} +0 & 0 & 1 & 0\\ +0 & 1 & 0 & 0\\ +-1 & 0 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + = \left[\begin{array}{cccc} +0 & -1 & 0 & 0\\ +0 & 0 & 1 & 0\\ +-1 & 0 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.43)$$ \noindent and the point {\bf u} transforms into {\bf w} as -$$\left[\matrix{-3\cr - 2\cr - -7\cr - 1} - \right] - = \left[\matrix{ 0 & -1 & 0 & 0\cr - 0 & 0 & 1 & 0\cr - -1 & 0 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right] - \left[\matrix{7\cr - 3\cr - 2\cr - 1} - \right]\eqno(1.44)$$ +$$\left[\begin{array}{c} +-3\\ +2\\ +-7\\ +1 +\end{array}\right] + = \left[\begin{array}{cccc} +0 & -1 & 0 & 0\\ +0 & 0 & 1 & 0\\ +-1 & 0 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{c} +7\\ +3\\ +2\\ +1 +\end{array}\right]\eqno(1.44)$$ \noindent We should expect this, as matrix multiplication is noncommutative. @@ -35647,41 +35698,47 @@ $4{\bf i}-3{\bf j}+7{\bf k}$. We obtain the translation from Equation 1.27 and the rotation from Equation 1.41. The matrix expression is $${\bf Trans(4,-3,7)}{\bf Rot(y,90)}{\bf Rot(z,90)} - = \left[\matrix{1 & 0 & 0 & 4\cr - 0 & 1 & 0 & -3\cr - 0 & 0 & 1 & 7\cr - 0 & 0 & 0 & 1} - \right] - \left[\matrix{0 & 0 & 1 & 0\cr - 1 & 0 & 0 & 0\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right] - = \left[\matrix{0 & 0 & 1 & 4\cr - 1 & 0 & 0 & -3\cr - 0 & 1 & 0 & 7\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.46)$$ + = \left[\begin{array}{cccc} +1 & 0 & 0 & 4\\ +0 & 1 & 0 & -3\\ +0 & 0 & 1 & 7\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{cccc} +0 & 0 & 1 & 0\\ +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + = \left[\begin{array}{cccc} +0 & 0 & 1 & 4\\ +1 & 0 & 0 & -3\\ +0 & 1 & 0 & 7\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.46)$$ \noindent and our point ${\bf w} = 7{\bf i}+3{\bf j}+2{\bf k}$ transforms into {\bf x} as -$$\left[\matrix{ 6\cr - 4\cr - 10\cr - 1} - \right] - = \left[\matrix{0 & 0 & 1 & 4\cr - 1 & 0 & 0 & -3\cr - 0 & 1 & 0 & 7\cr - 0 & 0 & 0 & 1} - \right] - \left[\matrix{7\cr - 3\cr - 2\cr - 1} - \right]\eqno(1.47)$$ +$$\left[\begin{array}{c} +6\\ +4\\ +10\\ +1 +\end{array}\right] + = \left[\begin{array}{cccc} +0 & 0 & 1 & 4\\ +1 & 0 & 0 & -3\\ +0 & 1 & 0 & 7\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{c} +7\\ +3\\ +2\\ +1 +\end{array}\right]\eqno(1.47)$$ \subsection{Coordinate Frames} @@ -35692,21 +35749,24 @@ $[0,0,0,1]^{\rm T}$ lies at the origin of the second coordinate frame. Its transformation corresponds to the right hand column of the transformation matrix. Consider the transform in Equation 1.47 -$$\left[\matrix{ 4\cr - -3\cr - 7\cr - 1} - \right] - = \left[\matrix{0 & 0 & 1 & 4\cr - 1 & 0 & 0 & -3\cr - 0 & 1 & 0 & 7\cr - 0 & 0 & 0 & 1} - \right] - \left[\matrix{0\cr - 0\cr - 0\cr - 1} - \right]\eqno(1.48)$$ +$$\left[\begin{array}{c} +4\\ +-3\\ +7\\ +1 +\end{array}\right] + = \left[\begin{array}{cccc} +0 & 0 & 1 & 4\\ +1 & 0 & 0 & -3\\ +0 & 1 & 0 & 7\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{c} +0\\ +0\\ +0\\ +1 +\end{array}\right]\eqno(1.48)$$ \noindent The transform of the null vector is $[4,-3,7,1]^{\rm T}$, the right @@ -35737,11 +35797,12 @@ made with respect to the fixed reference coordinate frame. Thus, in the example given, $${\bf Trans(4,-3,7)}{\bf Rot(y,90)}{\bf Rot(z,90)} - = \left[\matrix{0 & 0 & 1 & 4\cr - 1 & 0 & 0 & -3\cr - 0 & 1 & 0 & 7\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.49)$$ + = \left[\begin{array}{cccc} +0 & 0 & 1 & 4\\ +1 & 0 & 0 & -3\\ +0 & 1 & 0 & 7\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.49)$$ \noindent the frame is first rotated around the reference $z$ axis by @@ -35766,41 +35827,47 @@ about the $z$ axis, and a translation of 10 units in the $x$ direction, we obtain a new position {\bf X} when the change is made in the base coordinates ${\bf X} = {\bf T} {\bf C}$ -$$\left[\matrix{0 & 0 & 1 & 0\cr - 1 & 0 & 0 & 20\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right] - = \left[\matrix{0 & -1 & 0 & 10\cr - 1 & 0 & 0 & 0\cr - 0 & 0 & 1 & 0\cr - 0 & 0 & 0 & 1} - \right] - \left[\matrix{1 & 0 & 0 & 20\cr - 0 & 0 & -1 & 10\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.50)$$ +$$\left[\begin{array}{cccc} +0 & 0 & 1 & 0\\ +1 & 0 & 0 & 20\\ +0 & 1 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + = \left[\begin{array}{cccc} +0 & -1 & 0 & 10\\ +1 & 0 & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{cccc} +1 & 0 & 0 & 20\\ +0 & 0 & -1 & 10\\ +0 & 1 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.50)$$ \noindent and a new position {\bf Y} when the change is made relative to the frame axes as ${\bf Y} = {\bf C} {\bf T}$ -$$\left[\matrix{0 & -1 & 0 & 30\cr - 0 & 0 & -1 & 10\cr - 1 & 0 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right] - = \left[\matrix{1 & 0 & 0 & 20\cr - 0 & 0 & -1 & 10\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right] - \left[\matrix{0 & -1 & 0 & 10\cr - 1 & 0 & 0 & 0\cr - 0 & 0 & 1 & 0\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.51)$$ +$$\left[\begin{array}{cccc} +0 & -1 & 0 & 30\\ +0 & 0 & -1 & 10\\ +1 & 0 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + = \left[\begin{array}{cccc} +1 & 0 & 0 & 20\\ +0 & 0 & -1 & 10\\ +0 & 1 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{cccc} +0 & -1 & 0 & 10\\ +1 & 0 & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.51)$$ \subsection{Objects} @@ -35814,11 +35881,12 @@ $90^\circ$ about the $y$ axis, followed by a translation of four units in the $x$ direction, we can describe the transformation as $${\bf Trans(4,0,0)}{\bf Rot(y,90)}{\bf Rot(z,90)} = - \left[\matrix{0 & 0 & 1 & 4\cr - 1 & 0 & 0 & 0\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.52)$$ + \left[\begin{array}{cccc} +0 & 0 & 1 & 4\\ +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.52)$$ \noindent The transformation matrix represents the operation of rotation and @@ -35826,21 +35894,24 @@ translation on a coordinate frame originally aligned with the reference coordinate frame. We may transform the six points of the object as -$$\left[\matrix{4 & 4 & 6 & 6 & 4 & 4\cr - 1 & -1 & -1 & 1 & 1 & -1\cr - 0 & 0 & 0 & 0 & 4 & 4\cr - 1 & 1 & 1 & 1 & 1 & 1} - \right] - = \left[\matrix{0 & 0 & 1 & 4\cr - 1 & 0 & 0 & 0\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right] - \left[\matrix{1 & -1 & -1 & 1 & 1 & -1\cr - 0 & 0 & 0 & 0 & 4 & 4\cr - 0 & 0 & 2 & 2 & 0 & 0\cr - 1 & 1 & 1 & 1 & 1 & 1} - \right]\eqno(1.53)$$ +$$\left[\begin{array}{cccccc} +4 & 4 & 6 & 6 & 4 & 4\\ +1 & -1 & -1 & 1 & 1 & -1\\ +0 & 0 & 0 & 0 & 4 & 4\\ +1 & 1 & 1 & 1 & 1 & 1 +\end{array}\right] + = \left[\begin{array}{cccc} +0 & 0 & 1 & 4\\ +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{cccccc} +1 & -1 & -1 & 1 & 1 & -1\\ +0 & 0 & 0 & 0 & 4 & 4\\ +0 & 0 & 2 & 2 & 0 & 0\\ +1 & 1 & 1 & 1 & 1 & 1 +\end{array}\right]\eqno(1.53)$$ It can be seen that the object described bears the same fixed relationship to its coordinate frame, whose position and orientation @@ -35866,50 +35937,56 @@ $[1,0,0,0]^{\rm T}$ and $[0,1,0,0]^{\rm T}$, respectively. The location of the origin is $[0,0,-4,1]^{\rm T}$ with respect to the transformed frame and thus the inverse transformation is -$${\bf T^{-1}} = \left[\matrix{0 & 1 & 0 & 0\cr - 0 & 0 & 1 & 0\cr - 1 & 0 & 0 & -4\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.54)$$ +$${\bf T^{-1}} = \left[\begin{array}{cccc} +0 & 1 & 0 & 0\\ +0 & 0 & 1 & 0\\ +1 & 0 & 0 & -4\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.54)$$ \noindent That this is indeed the tranform inverse is easily verifyed by multiplying it by the transform {\bf T} to obtain the identity transform -$$\left[\matrix{1 & 0 & 0 & 0\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 1 & 0\cr - 0 & 0 & 0 & 1} - \right] - = \left[\matrix{0 & 1 & 0 & 0\cr - 0 & 0 & 1 & 0\cr - 1 & 0 & 0 & -4\cr - 0 & 0 & 0 & 1} - \right] - \left[\matrix{0 & 0 & 1 & 4\cr - 1 & 0 & 0 & 0\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.55)$$ +$$\left[\begin{array}{cccc} +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + = \left[\begin{array}{cccc} +0 & 1 & 0 & 0\\ +0 & 0 & 1 & 0\\ +1 & 0 & 0 & -4\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{cccc} +0 & 0 & 1 & 4\\ +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.55)$$ \noindent In general, given a transform with elements -$${\bf T} = \left[\matrix{n_x & o_x & a_x & p_x\cr - n_y & o_y & a_y & p_y\cr - n_z & o_z & a_z & p_z\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.56)$$ +$${\bf T} = \left[\begin{array}{cccc} +n_x & o_x & a_x & p_x\\ +n_y & o_y & a_y & p_y\\ +n_z & o_z & a_z & p_z\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.56)$$ \noindent then the inverse is -$${\bf T^{-1}} = \left[\matrix{n_x & n_y & n_z & -{\bf p} \cdot {\bf n}\cr - o_x & o_y & o_z & -{\bf p} \cdot {\bf o}\cr - a_x & a_y & a_z & -{\bf p} \cdot {\bf a}\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.57)$$ +$${\bf T^{-1}} = \left[\begin{array}{cccc} +n_x & n_y & n_z & -{\bf p} \cdot {\bf n}\\ +o_x & o_y & o_z & -{\bf p} \cdot {\bf o}\\ +a_x & a_y & a_z & -{\bf p} \cdot {\bf a}\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.57)$$ \noindent where {\bf p}, {\bf n}, {\bf o}, and {\bf a} are the four column @@ -35932,11 +36009,12 @@ around an arbitrary vector {\bf k} located at the origin. In order to do this we will imagine that {\bf k} is the $z$ axis unit vector of a coordinate frame {\bf C} -$${\bf C} = \left[\matrix{n_x & o_x & a_x & p_x\cr - n_y & o_y & a_y & p_y\cr - n_z & o_z & a_z & p_z\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.58)$$ +$${\bf C} = \left[\begin{array}{cccc} +n_x & o_x & a_x & p_x\\ +n_y & o_y & a_y & p_y\\ +n_z & o_z & a_z & p_z\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.58)$$ $${\bf k} = a_x{\bf i} + a_y{\bf j} + a_z{\bf k}\eqno(1.59)$$ @@ -35985,67 +36063,71 @@ Multiplying ${\bf Rot(z,\theta)}$ on the right by ${\bf C^{-1}}$ we obtain $${\bf Rot(z,\theta)} {\bf C^{-1}} - = \left[\matrix{cos \theta & -sin \theta & 0 & 0\cr - sin \theta & cos \theta & 0 & 0\cr - 0 & 0 & 1 & 0\cr - 0 & 0 & 0 & 1} - \right] - \left[\matrix{n_x & n_y & n_z & 0\cr - o_x & o_x & o_z & 0\cr - a_x & a_y & a_z & 0\cr - 0 & 0 & 0 & 1} - \right]$$ - -$$ = \left[\matrix{n_x cos \theta - o_x sin \theta & - n_y cos \theta - o_y sin \theta & - n_z cos \theta - o_z sin \theta & 0\cr - n_x sin \theta + o_x cos \theta & - n_y sin \theta + o_y cos \theta & - n_z sin \theta + o_z cos \theta & 0\cr - a_x & a_y & a_z & 0\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.66)$$ + = \left[\begin{array}{cccc} +cos \theta & -sin \theta & 0 & 0\\ +sin \theta & cos \theta & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{cccc} +n_x & n_y & n_z & 0\\ +o_x & o_x & o_z & 0\\ +a_x & a_y & a_z & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]$$ + +$$ = \left[\begin{array}{cccc} +n_x cos \theta - o_x sin \theta & +n_y cos \theta - o_y sin \theta & +n_z cos \theta - o_z sin \theta & 0\\ +n_x sin \theta + o_x cos \theta & +n_y sin \theta + o_y cos \theta & +n_z sin \theta + o_z cos \theta & 0\\ +a_x & a_y & a_z & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.66)$$ \noindent premultiplying by -$${\bf C} = \left[\matrix{n_x & o_x & a_x & 0\cr - n_y & o_y & a_y & 0\cr - n_z & o_z & a_z & 0\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.67)$$ +$${\bf C} = \left[\begin{array}{cccc} +n_x & o_x & a_x & 0\\ +n_y & o_y & a_y & 0\\ +n_z & o_z & a_z & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.67)$$ \noindent we obtain ${\bf C} {\bf Rot(z,\theta)} {\bf C^{-1}}$ -$$\left[\matrix{ +$$\left[\begin{array}{c} n_x n_x cos \theta - n_x o_x sin \theta + n_x o_x sin \theta + o_x o_x -cos \theta + a_x a_x\cr +cos \theta + a_x a_x\\ n_y n_x cos \theta - n_y o_x sin \theta + n_x o_y sin \theta + o_x o_y -cos \theta + a_y a_x\cr +cos \theta + a_y a_x\\ n_z n_x cos \theta - n_z o_x sin \theta + n_x o_z sin \theta + o_x o_z -cos \theta + a_z a_x\cr -0} -\right.$$ +cos \theta + a_z a_x\\ +0 +\end{array}\right.$$ -$$\matrix{ +$$\begin{array}{c} n_x n_y cos \theta - n_x o_y sin \theta + n_y o_x sin \theta + o_y o_x -cos \theta + a_x a_y\cr +cos \theta + a_x a_y\\ n_y n_y cos \theta - n_y o_y sin \theta + n_y o_y sin \theta + o_y o_y -cos \theta + a_y a_y\cr +cos \theta + a_y a_y\\ n_z n_y cos \theta - n_z o_y sin \theta + n_y o_z sin \theta + o_y o_z -cos \theta + a_z a_y\cr -0}\eqno(1.68)$$ +cos \theta + a_z a_y\\ +0\end{array}\eqno(1.68)$$ -$$\left.\matrix{ +$$\left.\begin{array}{cc} n_x n_z cos \theta - n_x o_z sin \theta + n_z o_x sin \theta + o_z o_x -cos \theta + a_x a_x & 0\cr +cos \theta + a_x a_x & 0\\ n_y n_z cos \theta - n_y o_z sin \theta + n_z o_y sin \theta + o_z o_y -cos \theta + a_y a_z & 0\cr +cos \theta + a_y a_z & 0\\ n_z n_z cos \theta - n_z o_z sin \theta + n_z o_z sin \theta + o_z o_z -cos \theta + a_z a_z & 0\cr -0 & 1} -\right]$$ +cos \theta + a_z a_z & 0\\ +0 & 1 +\end{array}\right]$$ \noindent Simplifying, using the following relationships:\\ @@ -36069,21 +36151,21 @@ the versine, abbreviated ${\bf vers \ \theta}$, is defined as ${\bf vers \ \theta} = (1 - cos \ \theta)$, ${k_x = a_x}$, ${k_y = a_y}$ and ${k_z = a_z}$. We obtain ${\bf Rot(k,\theta)} =$ -$$\left[\matrix{ +$$\left[\begin{array}{cccc} k_x k_x vers \theta + cos \theta & k_y k_x vers \theta - k_z sin \theta & k_z k_x vers \theta + k_y sin \theta & -0\cr +0\\ k_x k_y vers \theta + k_z sin \theta & k_y k_y vers \theta + cos \theta & k_z k_y vers \theta - k_x sin \theta & -0\cr +0\\ k_x k_z vers \theta - k_y sin \theta & k_y k_z vers \theta + k_x sin \theta & k_z k_z vers \theta + cos \theta & -0\cr -0 & 0 & 0 & 1} -\right]\eqno(1.70)$$ +0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.70)$$ \noindent This is an important result and should be thoroughly understood before @@ -36096,11 +36178,12 @@ ${k_z = 0}$. Substituting these values of {\bf k} into Equation 1.70 we obtain $${\bf Rot(x,\theta)} = -\left[\matrix{1 & 0 & 0 & 0\cr - 0 & cos \theta & -sin \theta & 0\cr - 0 & sin \theta & cos \theta & 0\cr - 0 & 0 & 0 & 1} -\right]\eqno(1.71)$$ +\left[\begin{array}{cccc} +1 & 0 & 0 & 0\\ +0 & cos \theta & -sin \theta & 0\\ +0 & sin \theta & cos \theta & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.71)$$ \noindent as before. @@ -36113,35 +36196,37 @@ Given any arbitrary rotational transformation, we can use Equation made as follows. Given a rotational transformation {\bf R} $${\bf R} = -\left[\matrix{n_x & o_x & a_x & 0\cr - n_y & o_y & a_y & 0\cr - n_z & o_z & a_z & 0\cr - 0 & 0 & 0 & 1} -\right]\eqno(1.72)$$ +\left[\begin{array}{cccc} +n_x & o_x & a_x & 0\\ +n_y & o_y & a_y & 0\\ +n_z & o_z & a_z & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.72)$$ \noindent we may equate {\bf R} to {\bf Rot(k,$\theta$)} -$$\left[\matrix{n_x & o_x & a_x & 0\cr - n_y & o_y & a_y & 0\cr - n_z & o_z & a_z & 0\cr - 0 & 0 & 0 & 1} - \right] = $$ -$$\left[\matrix{ +$$\left[\begin{array}{cccc} +n_x & o_x & a_x & 0\\ +n_y & o_y & a_y & 0\\ +n_z & o_z & a_z & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] = $$ +$$\left[\begin{array}{cccc} k_x k_x vers \theta + cos \theta & k_y k_x vers \theta - k_z sin \theta & k_z k_x vers \theta + k_y sin \theta & -0\cr +0\\ k_x k_y vers \theta + k_z sin \theta & k_y k_y vers \theta + cos \theta & k_z k_y vers \theta - k_x sin \theta & -0\cr +0\\ k_x k_z vers \theta - k_y sin \theta & k_y k_z vers \theta + k_x sin \theta & k_z k_z vers \theta + cos \theta & -0\cr -0 & 0 & 0 & 1} -\right]\eqno(1.73)$$ +0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.73)$$ \noindent Summing the diagonal terms of Equation 1.73 we obtain @@ -36151,10 +36236,11 @@ k_x^2 vers \theta + cos \theta + k_y^2 vers \theta + cos \theta + k_z^2 vers \theta + cos \theta + 1\eqno(1.74)$$ -$$\left.\matrix{ n_x+o_y+a_z & = & - (k_x^2+k_y^2+k_z^2)vers \theta + 3 cos \theta\cr - & = & 1 + 2 cos \theta} - \right.\eqno(1.75)$$ +$$\left.\begin{array}{ccccc} +n_x+o_y+a_z & = & + (k_x^2+k_y^2+k_z^2)vers \theta + 3 cos \theta\\ + & = & 1 + 2 cos \theta +\end{array} \right.\eqno(1.75)$$ \noindent and the cosine of the angle of rotation is @@ -36302,11 +36388,12 @@ Determine the equivalent axis and angle of rotation for the matrix given in Equations 1.41 $${\bf Rot(y,90)}{\bf Rot(z,90)} - = \left[\matrix{0 & 0 & 1 & 0\cr - 1 & 0 & 0 & 0\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.104)$$ + = \left[\begin{array}{cccc} +0 & 0 & 1 & 0\\ +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.104)$$ \noindent We first determine ${\bf cos \ \theta}$ from Equation 1.76 @@ -36367,11 +36454,12 @@ that we will make use of later. A transform {\bf T} -$${\bf T} = \left[\matrix{a & 0 & 0 & 0\cr - 0 & b & 0 & 0\cr - 0 & 0 & c & 0\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.113)$$ +$${\bf T} = \left[\begin{array}{cccc} +a & 0 & 0 & 0\\ +0 & b & 0 & 0\\ +0 & 0 & c & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.113)$$ \noindent will stretch objects uniformly along the $x$ axis by a factor $a$, @@ -36379,21 +36467,24 @@ along the $y$ axis by a factor $b$, and along the $z$ axis by a factor $c$. Consider any point on an object $x{\bf i}+y{\bf j}+z{\bf k}$; its tranform is -$$\left[\matrix{ax\cr - by\cr - cz\cr - 1} - \right] - = \left[\matrix{a & 0 & 0 & 0\cr - 0 & b & 0 & 0\cr - 0 & 0 & c & 0\cr - 0 & 0 & 0 & 1} - \right] - \left[\matrix{x\cr - y\cr - z\cr - 1} - \right]\eqno(1.114)$$ +$$\left[\begin{array}{c} +ax\\ +by\\ +cz\\ +1 +\end{array}\right] + = \left[\begin{array}{cccc} +a & 0 & 0 & 0\\ +0 & b & 0 & 0\\ +0 & 0 & c & 0\\ +0 & 0 & 0 & 1 +\end{array}\right] + \left[\begin{array}{c} +x\\ +y\\ +z\\ +1 +\end{array}\right]\eqno(1.114)$$ \noindent indicating stretching as stated. Thus a cube could be transformed into @@ -36412,11 +36503,12 @@ The Axiom code to perform this scale change is: \noindent The transform {\bf S} where -$${\bf S} = \left[\matrix{s & 0 & 0 & 0\cr - 0 & s & 0 & 0\cr - 0 & 0 & s & 0\cr - 0 & 0 & 0 & 1} - \right]\eqno(1.115)$$ +$${\bf S} = \left[\begin{array}{cccc} +s & 0 & 0 & 0\\ +0 & s & 0 & 0\\ +0 & 0 & s & 0\\ +0 & 0 & 0 & 1 +\end{array}\right]\eqno(1.115)$$ \noindent will scale any object by the factor $s$. @@ -36496,30 +36588,34 @@ $$y^\prime = {{y}\over{(1-{{y}\over{f}})}}\eqno(1.126)$$ The homogeneous transformation {\bf P} which produces the same result is -$${\bf P} = \left[\matrix{1 & 0 & 0 & 0\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 1 & 0\cr - 0 & -{{1}\over{f}} & 0 & 1} - \right]\eqno(1.127)$$ +$${\bf P} = \left[\begin{array}{cccc} +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & -{{1}\over{f}} & 0 & 1 +\end{array}\right]\eqno(1.127)$$ \noindent as any point $x{\bf i}+y{\bf j}+z{\bf k}$ transforms as -$$\left[\matrix{x\cr - y\cr - z\cr - {1 - {{{y}\over{f}}}}} - \right] - = \left[\matrix{1 & 0 & 0 & 0\cr - 0 & 1 & 0 & 0\cr - 0 & 0 & 1 & 0\cr - 0 & -{{1}\over{f}} & 0 & 1} - \right] - \left[\matrix{x\cr - y\cr - z\cr - 1} - \right]\eqno(1.128)$$ +$$\left[\begin{array}{c} +x\\ +y\\ +z\\ +{1 - {{{y}\over{f}}}} +\end{array}\right] + = \left[\begin{array}{cccc} +1 & 0 & 0 & 0\\ +0 & 1 & 0 & 0\\ +0 & 0 & 1 & 0\\ +0 & -{{1}\over{f}} & 0 & 1 +\end{array}\right] + \left[\begin{array}{c} +x\\ +y\\ +z\\ +1 +\end{array}\right]\eqno(1.128)$$ \noindent The image point $x^\prime$, $y^\prime$,, $z^\prime$, obtained by diff --git a/books/bookvol10.5.pamphlet b/books/bookvol10.5.pamphlet index 718d3d1..64a7b7f 100644 --- a/books/bookvol10.5.pamphlet +++ b/books/bookvol10.5.pamphlet @@ -3,6 +3,8 @@ \input{bookheader.tex} \mainmatter \setcounter{secnumdepth}{0} % override the one in bookheader.tex +\setcounter{tocdepth}{5} +\setcounter{secnumdepth}{5} \setcounter{chapter}{0} % Chapter 1 \chapter{Numerical Analysis} We can describe each number as $x^{*}$ which has a machine-representable @@ -76,6 +78,1825 @@ errors they introduce. Every effort will be made to minimize these errors. In particular, we will appeal to the machine generated code to see what approximations actually occur. +\chapter{The Quality of Computed Solutions by Sven Hammarling} +This is quoted with permission from Hammarling\cite{Hamm05} + +\section{Introduction} + +This report is concerned with the quality of the computed +numerical solutions of mathematical problems. For example, suppose +we wish to solve the system of linear equations $Ax=b$ using a numerical +software package. The package will return a computed solution, say +$\tilde{x}$, and we wish to judge whether or not $\tilde{x}$ is a +reasonable solution to the equations. Sadly, all too often software +packages return poor, or even incorrect, numerical results and give +the user no means by which to judge the quality of the numerical results. +In 1971, Leslie Fox commented [\cite{Foxx71} 1971, p. 296] + +\begin{quote} +“I have little doubt that about 80 per cent. of all the results printed +from the computer are in error to a much greater extent than the user +would believe, ...” +\end{quote} + +More than thirty years on that paper is still very relevant and worth reading. +Another very readable article is \cite{Fors70}. + +The quality of computed solutions is concerned with assessing how good +a computed solution is in some appropriate measure. Quality software +should implement reliable algorithms and should, if possible, provide +measures of solution quality. + +In this report we give an introduction to ideas that are important in +understanding and measuring the quality of computed solutions. In +particular we review the ideas of condition, stability and error +analysis, and their realisation in numerical software. We take as the +principal example LAPACK \cite{Ande99}, a package for the +solution of dense and banded linear algebra problems, but also draw +on examples from the NAG Library \cite{NAGa05} and elsewhere . The aim is +not to show how to perform an error analysis, but to indicate why an +understanding of the ideas is important in judging the quality of +numerical solutions, and to encourage the use of software that +returns indicators of the quality of the results. We give simple +examples to illustrate some of the ideas that are important when +designing reliable numerical software. + +Computing machines use floating point arithmetic for their +computation, and so we start with an introduction to floating point +numbers. + +\section{Floating Point Numbers and IEEE Arithmetic} + +Floating point numbers are a subset of the real numbers that can be +conveniently represented in the finite word length of a computer, +without unduly restricting the range of numbers represented. For +example, the ANSI/IEEE standard for binary floating point arithmetic +\cite{IEEE85} uses 64 bits to represent double precision numbers in the +approximate range $10^{\pm 308}$. A floating point number, $x$, can be +represented in terms of four integers as +\[x=\pm m \times b^{e-t}\] +where $b$ is the {\sl base} or {\sl radix}, $t$ is the {\sl precision}, +$e$ is the {\sl exponent} with an {\sl exponent range} of +$[e_{min},e_{max}]$ and $m$ is the {\sl mantissa} or {\sl significand}, +satisfying $0 \le m \le b^t-1$. If $x \ne 0$ and $m \ge b^{t-1}$ then +$x$ is said to be {\sl normalized}. An alternative, equivalent representation +of $x$ is +\[\begin{array}{rcl} +x&=& \pm 0.d_1d_2\ldots d_t \times b^e\\ + &=& \pm\left(\frac{d_1}{b}+\frac{d_2}{b^2}+\cdots+\frac{d_t}{b^t}\right) +\times b^e +\end{array}\] +where each digit satisfies $0 \le d_i \le b - 1$. If $d_1 \ne 0$ then we +see that $x$ is normalized. If $d_1 = 0$ and $x \ne 0$ then $x$ is said +to be {\sl denormalized}. Denormalized numbers between 0 and the smallest +normalized number are called {\sl subnormal}. Note that denormalized +numbers do not have the full $t$ digits of precision. + +The following example, which is not intended to be realistic, illustrates +the model. + +\subsection{Example 2.1 (Floating point numbers)} + +\[ b=2,\quad t=2,\quad e_{\rm min} = -2,\quad e_{\rm max}=2\] +{\sl All the normalized numbers have $d_1=1$ and either $d_2=0$ or $d_2=1$, +that is m is one of the two binary integers $m=10~(=2)$ or $m=11~(=3)$. +Denormalized numbers have $m=01~(=1)$. Thus the smallest positive normalized +number is $2 \times 2^{e_{\rm min}-t} = \frac{1}{8}$ and the largest is +$3 \times 2^{e_{\rm max}-t}=3$. The value +$1 \times 2^{e_{\rm min}-t}=\frac{1}{16}$ is the only positive subnormal +number in this system. The complete set of non-negative normalized numbers is +\[0,\frac{1}{8},\frac{3}{16},\frac{1}{4},\frac{3}{8},\frac{1}{2},\frac{3}{4}, +1,\frac{3}{2},2,3\] +The set of non-negative floating point numbers is illustrated in Figure 1, +where the subnormal number is indicated by a dashed line.} + +\begin{center} +\includegraphics[scale=0.75]{ps/v105hammfig1.eps}\\ +{\bf Figure 1: Floating Point Number Example} +\end{center} + +Note that floating point numbers are not equally spaced absolutely, +but the relative spacing between numbers is approximately equal. The +value +\[u = \frac{1}{2} \times b^{1-t}\eqno{(1)}\] +is called the {\sl unit roundoff}, or the {\sl relative machine precision} +and is the furthest distance relative to unity between a real number and the +nearest floating point number. In Example 2.1, $u = \frac{1}{4} = 0.25$ +and we can see, for example, that the furthest real number from 1.0 is +1.25 and the furthest real number from 2.0 is 2.5. $u$ is fundamental +to floating point error analysis. + +The value $e_M = 2u$ is called {\sl machine epsilon}. + +The ANSI/IEEE standard mentioned above (usually referred to as IEEE +arithmetic), which of course has $b = 2$, specifies: +\begin{itemize} +\item floating point number formats +\item results of the basic floating point operations +\item rounding modes +\item signed zero, infinity ($\pm\infty$) and not-a-number (NaN) +\item floating point exceptions and their handling and +\item conversion between formats +\end{itemize} + +Thankfully, nowadays almost all machines use IEEE arithmetic. There is +also a generic ANSI/IEEE, base independent, standard \cite{IEEE87}. The +formats supported by the ANSI/IEEE binary standard are indicated in +Table 1. +\vskip 0.25cm +\begin{tabular}{|l|l|l|l|l|} +\hline +Format & Precision & Exponent & Approx Range & Approx Precision\\ +\hline +Single & 24 bits & 8 bits & $10^{\pm 38}$ & $10^{-8}$\\ +Double & 53 bits & 11 bits & $10^{\pm 308}$ & $10^{-16}$\\ +Extended & $\ge$ 64 bits & $\ge$ 15 bits & $10^{\pm 4932}$ & $10^{-20}$\\ +\hline +\end{tabular} +\vskip 0.25cm +The default rounding mode for IEEE arithmetic is round to nearest, in +which a real number is represented by the nearest floating point +number, with rules as to how to handle a tie [\cite{Over01}, Chapter 5]. + +Whilst the ANSI/IEEE standard has been an enormous help in +standardizing floating point computation, it should be noted that +moving a computation between machines that implement IEEE arithmetic +does not guarantee that the computed results will be the +same. Variations can occur due to such things as compiler +optimization, the use of extended precision registers, and fused +multiply-add. + +Further discussion of floating point numbers and IEEE arithmetic can +be found in \cite{High02} and \cite{Over01}. + +The value $u$ can be obtained from the LAPACK function SLAMCH, for +single precision arithmetic, or DLAMCH for double precision arithmetic +by calling the function with the argument CMACH as ’e’, and is also +returned by the NAG Fortran Library routine +X02AJF\footnote{In some ports it actually returns $u+b^{1=2t}$. See +the X02 Chapter introduction\cite{NAGb05}.} It should be +noted that on machines that truncate, rather than round, $M$ is returned +in place of $u$, but such machines are now rare. It should also be noted +that ’e’ in S/DLAMCH represents eps, but this should not be +confused with $M$. The Matlab built in variable eps returns $M$\cite{Math05}, +as does the Fortran 95/Fortran 2003 numeric enquiry function epsilon +\cite{Metc96,Metc04}. + +\section{Why Worry about Computed Solutions?} + +In this section we consider some simple examples of numerical +computation that need care in order to obtain reasonable +solutions. For clarity of exposition, most of the examples in this and +the following sections are illustrated using decimal floating point +(significant figure) arithmetic, with round to nearest. + +The first example illustrates the problem of damaging subtraction, usually +referred to as {\sl cancellation}. + +\subsection{Example 3.1 (Cancellation)} + +{\sl Using four figure decimal arithmetic, suppose we wish to compute +$s=1.000+1.000\times 10^4 - 1.000\times 10^4$. If we compute in the standard +way from left to right we obtain} +\[\begin{array}{rcl} +s & = & 1.000+1.000\times 10^4-1.000\times 10^4\\ + & = & (1.000 + 1.000\times 10^4) - 1.000\times 10^4\\ + & = & 1.000\times 10^4 - 1.000\times 10^4 \\ + & = & 0 +\end{array}\] + +{\sl instead of the correct result of 1.0. Although the cancellation +(subtraction) was performed exactly, it lost all the information for +the solution.} + +As Example 3.1 illustrates, the cause of the poor result often happens +before the cancellation, and the cancellation is just the final nail +in the coffin. In Example 3.1, the damage was done in computing +$s = 1.000 + 1.000\times 10^4$ , where we lost important information (1.000). +It should be said that the subtraction of nearly equal numbers is not +always damaging. + +Most problems have alternative formulations which are theoretically +equivalent, but may computationally yield quite different results. The +following example illustrates this in the case of computing sample +variances. + +\subsection{Example 3.2 (Sample variance \cite{High02},Section 1.9)} + +{\sl The sample variance of a set of n values $x_1,x_2,\ldots,x_n$ is +defined as} +\[s_n^2 = \frac{1}{n-1}\sum_{i=1}^n{(x_i-\overline{x})^2}\eqno{(2)}\] +{\sl where $\overline{x}$ is the sample mean of the n values} +\[\overline{x} = \frac{1}{n}\sum_{i=1}^n{x_i}\] + +{\sl An alternative, theoretically equivalent, formula to compute the +sample variance which requires only one pass through the data is given by} +\[s_n^2=\frac{1}{n-1}\left(\sum_{i=1}^n{x_i^2}- +\frac{1}{n}\left(\sum_{i=1}^n{x_i}\right)^2\right)\eqno{(3)}\] + +{\sl If $x^T$ = (10000 10001 10002) then using 8 figure arithmetic (2) +gives $s^2 = 1.0$, the correct answer, but (3) gives $s^2 = 0.0$, +with a relative error of 1.0.} + +(3) can clearly suffer from cancellation, as illustrated in the example. +On the other hand, (2) always gives good results unless $n$ is very large +[\cite{High02}, Problem 1.10]. See also \cite{Chan83} +for further discussion of the problem of computing sample variances. + +Sadly, it is not unknown for software packages and calculators to +implement the algorithm of (3). For example in Excel 2002 from +Microsoft Office XP (and in previous versions of Excel also), the +function STDEV computes the standard deviation, $s$, of the data +\[x^T = (100000000 100000001 100000002)\] +as $s = 0$. Considering the pervasive use of Excel and the importance +of standard deviation and its use in applications, it is disappointing +to realise that (3) has been used by these versions of +Excel.\footnote{The algorithm has at last been replaced in Excel from +Office 2003, which now gives the correct answer.} See +\cite{Coxx00} for further information, as well as \cite{Knus98}, +\cite{Mccu99} and \cite{Mccu02}. The +spreadsheet from OpenOffice.org version 1.0.2 produces the same +result, but gives no information on the method used in its help +system; on the other hand the Gnumeric spreadsheet (version 1.0.12) +gives the correct result, although again the function description does +not describe the method used.\footnote{OpenOffice.org version 2.0 also +now gives the correct result} + +A result that is larger than the largest representable floating point +number is said to {\sl overflow}. For example, in double precision IEEE +arithmetic for which the approximate range is $10^{\pm 308}$, +if $x = 10^{200}$, then $x^2$ would overflow. Similarly, +$x^{-2}$ is said to {\sl underflow} because it is smaller than the smallest +non-zero representable floating point number. + +As with the unit rounding error or machine epsilon discussed in +Section 2, the overflow and underflow thresholds can be obtained +from the LAPACK function S/DLAMCH by calling the function with the +argument CMACH as ’o’ and ’u’ respectively; from the NAG Fortran +Library routines X02ALF and X02AKF respectively; the Matlab built in +variables {\tt realmax} and {\tt realmin}; and from the Fortran 95 +numeric enquiry functions {\tt huge} and {\tt tiny}. + +Care needs to be taken to avoid unnecessary overflow and damaging +underflow. The following example illustrates this care in the case of +computing the hypotenuse of the right angled triangle shown in Figure 2. + +\begin{center} +\includegraphics{ps/v105hammfig2.eps}\\ +{\bf Figure 2: Hypotenuse of a right angled triangle} +\end{center} + +\subsection{Example 3.3 (Hypotenuse)} + +{\sl in Figure 2, if $x$ or $y$ is very large there is a danger of +overflow, even if $z$ is representable. Assuming that $x$ and $y$ are +non-negative, a safe method of computing $z$ is} +\[a={\rm max}(x,y),~ b={\rm min}(x,y)\] +\[z= +\begin{cases} +a\sqrt{1+\displaystyle\left(\frac{b}{a}\right)^2},&\mbox{}{a > 0}\\ +0&\mbox{}{a=0} +\end{cases}\] + +{\sl This also avoids $z$ being computed as zero if $x^2$ and $y^2$ both +underflow. We note that \cite{Stew98}, p.139 and p144 +actually recommends computing z as} +\[z= +\begin{cases} +s\sqrt{\displaystyle\left(\frac{x}{s}\right)^2+ +\displaystyle\left(\frac{y}{s}\right)^2}, +{\rm where\ }s=x+y,&\mbox{}x>0\\ +0&\mbox{}s=0 +\end{cases}\] +{\sl because this is slightly more accurate on a hexadecimal machine. An +interesting alternative for computing Pythagorean sums is given in +\cite{Mole83}; see also \cite{Dubr83} +and \cite{High05}, section 22.9.} + +We can see that (3) of Example 3.2 also has the potential for overflow +and underflow and, as well as implementing this formula rather than a +more stable version, Excel does not take the necessary care to avoid +underflow and overflow. For example, for the values (1.0E200, +1.0E200), STDEV in Excel 2003 from Microsoft Office 2003 returns the +mysterious symbol \#NUM!, which signifies a numeric exception, in this +case overflow, due to the fact that $(10.0^{200})^2$ overflows in IEEE +double precision arithmetic. The correct standard deviation is of +course 0. Similarily, for the values (0, 1.0E-200, 2.0E-200), STDEV +returns the value 0 rather than the correct value of +1.0E-200. OpenOffice.org version 1.0.2 also returns zero for this +data, and overflows on the previous data. Mimicking Excel is not +necessarily a good thing!\footnote{OpenOffice.org version 2.0 now +produces the correct result for the data (1.0E200,1.0E200), but +underflows for (0,1.0E-200,2.0E-200)} + +The computation of the modulus of a complex number $x = x_r + ix_i$ +requires almost the same computation as that in Example 3.3. + +\subsection{Example 3.4 (Modulus of a complex number)} +\[|x|=\sqrt{x_r^2+x_i^2}\] +\[a={\rm max}(|x_r|,|x_i|),~b={\rm min}(|x_r|,|x_i|)\] +\[|x|= +\begin{cases} +a\sqrt{1+\displaystyle\left(\frac{b}{a}\right)^2}&\mbox{}a>0\\ +0&\mbox{}a=0 +\end{cases}\] +{\sl Again this also avoids $|x|$ being computed as zero if +$x_r^2$ and $x_i^2$ both underflow.} + +Another example where care is needed in complex arithmetic is complex +division +\[\frac{x}{y} = \frac{x_r+ix_i}{y_r+iy_i} +=\frac{(x_r+ix_i)(y_r-iy_i)}{y_r^2+y_i^2}\] + +Again, some scaling is required to avoid overflow and underflow. See +for example \cite{Smit62}, \cite{Stew85} and +\cite{Prie04}. Algol 60 +procedures for the complex operations of modulus, division and +square root are given in \cite{Mart68} and the NAG +Library Chapter, A02, for complex arithmetic has routines based upon +those Algol procedures, see for example \cite{NAGb05}. +A careful C function +is given in the Priest reference above. Occasionally, some aspect of +complex floating point arithmetic is incorrectly implemented, see for +example \cite{Blac97}, Section 7. + +Another example, similar to the previous examples, requiring care to +avoid overflow and damaging underflow is that of real plane rotations +where we need to compute $c = \cos\theta$ and $s = \sin\theta$ such that +\[c=\frac{x}{z},~s=\frac{y}{z},~{\rm where\ }z=\sqrt{x^2+y^2}\] +or alternatively +\[c=\frac{-x}{z},~s=\frac{-y}{z}\] + +Another convenient way to express the two choices is as +\[c=\frac{\pm 1}{\sqrt{1+t^2}},~s=ct, +~{\rm where\ }t\equiv\tan\theta=\frac{x}{y}\eqno{(4)}\] + +If $G$ is the {\sl plane rotation matrix} +\[G=\left( +\begin{array}{cc} +c & s\\ +-s & c +\end{array} +\right)\] + +then, with the choices of (4), +\[G\left(\begin{array}{c} +x\\ +y +\end{array}\right)= +\left(\begin{array}{c} +\pm z\\ +0 +\end{array}\right)\] + +When used in this way for the introduction of zeros the rotation is +generally termed a Givens plane rotation \cite{Give54}; \cite{Golu96}. +Givens himself certainly took care in the computation +of $c$ and $s$. To see an extreme case of the detailed consideration +necessary to implement a seemingly simple algorithm, but to be +efficient, to preserve as much accuracy as possible throughout the +range of floating point numbers, and to avoid overflow and damaging +underflow see \cite{Bind02}, where the computation of the +Givens plane rotation is fully discussed. + +Sometimes computed solutions are in some sense reasonable, but may not +be what the user was expecting. In the next example, the computed +solution is close to the exact solution, but does not meet a +constraint that the user might have expected the solution to meet. + +\subsection{Example 3.5 (Sample mean)\cite{High98}} + +{\sl Using three figure floating point decimal arithmetic:} +\[(5.01+5.03)/2 \Rightarrow 10.0/2 \Rightarrow 5.00\] +{\sl and we see that the computed value is outside the range of the data, +although it is not inaccurate.} + +Whether or not such a result matters depends upon the application, but +is an issue that needs to be considered when implementing numerical +algorithms. For instance if +\[y=\cos x\] +then we probably expect the property $|y| \le 1$ to be preserved +computationally, so that a value $|y| > 1$ is never returned. For a +monotonic function we may expect monotonicity to be preserved computationally. + +In the next section we look at ideas that help our understanding of what +constitutes a quality solution. + +\section{Condition, Stability and Error Analysis} +\subsection{Condition} + +Firstly we look at the condition of a problem. The condition of a +problem is concerned with the sensitivity of the problem to +perturbations in the data. A problem is ill-conditioned if small +changes in the data cause relatively large changes in the +solution. Otherwise a problem is well-conditioned. Note that +condition is concerned with the sensitivity of the problem, and is +independent of the method we use to solve the problem. We now give +some examples to illustrate somewhat ill-conditioned problems. + +\subsubsection{Example 4.1 (Cubic equation)} + +{\sl Consider the cubic equation} +\[x^3-21x^2+120x-100=0\] +{\sl whose exact roots are $x_1=1$, $x_2=x_3=10$. If we perturb the +coefficient of $x^3$ to give} +\[0.99x^3-21x^2+120x-100=0\] +{\sl the roots become $x_1\approx 1.000$,$x_2\approx 11.17$, +$x_3\approx 9.041$, so that the changes in the two roots $x_2$ and +$x_3$ are significantly greater than the change in the coefficient. +On the other hand, the roots of the perturbed cubic equation} +\[1.01x^3-21x^2+120x-100=0\] +{\sl are $x_1\approx 1.000$,$x_2,x_3\approx 9.896\pm 1.044i$,and +this time the double root has become a complex conjugate pair with +a significant imaginary part.} + +{\sl We can see that the roots $x_2$ and $x_3$ are ill-conditionsed. +Note that we cannot deduce anything about the condition of $x_1$ just +from this data. The three cubic polynomials are plotted in Figure 3.} + +\begin{center} +\includegraphics{ps/v105hammfig3.eps}\\ +{\bf Figure 3: Cubic Equation Example} +\end{center} + +\subsubsection{Example 4.2 (Eigenvalue problem)} +{\sl The matrix} +\[A=\left(\begin{array}{cccc} +10 & 100 & 0 & 0\\ +0 & 10 & 100 & 0\\ +0 & 0 & 10 & 100\\ +0 & 0 & 0 & 10 +\end{array}\right)\] +{\sl has eigenvalues $\lambda_1=\lambda_2=\lambda_3=\lambda_4=10$, +whereas the slightly perturbed matrix} +\[B=\left(\begin{array}{cccc} +10 & 100 & 0 & 0\\ +0 & 10 & 100 & 0\\ +0 & 0 & 10 & 100\\ +10^{-6} & 0 & 0 & 10 +\end{array}\right)\] +{\sl has eigenvalues $\lambda_1=11$,$\lambda_2,\lambda_3=10\pm i$, +$\lambda_4=9$.} + +\subsubsection{Example 4.3 (Integral)} +\[I=\int_{-10}^{10}{(ae^x-be^{-x})~dx}\] +{\sl When $a=b=1$, $I=0$, but when $a=1$, $b+1.01$, $I\approx -220$. +The function $f(x)=xe^x-be^{-x}$, when $a=b=1$ is plotted in Figure 4. +Notice that the vertical scale has a scale factor of $10^4$, so that a +small change in function can make a large change in the area under the +curve.} + +\begin{center} +\includegraphics{ps/v105hammfig4.eps}\\ +{\bf Figure 4: Integral Example} +\end{center} + +\subsubsection{Example 4.4 (Linear Equations)} +{\sl The equations $Ax=b$ given by} +\[\left(\begin{array}{cc} +99 & 98\\ +100 & 99 +\end{array}\right) +\left(\begin{array}{c} +x_1\\ +x_2 +\end{array}\right)= +\left(\begin{array}{c} +197\\ +199 +\end{array}\right)\eqno{(5)}\] +{\sl have the solution $x_1=x+2=1$, but the equations} +\[\left(\begin{array}{cc} +98.99 & 98\\ +100 & 99 +\end{array}\right) +\left(\begin{array}{c} +x_1\\ +x_2 +\end{array}\right)= +\left(\begin{array}{c} +197\\ +199 +\end{array}\right)\] +{\sl have the solution $x_1=100$, $x_2=-99$. The two straight lines +represented by (5) are plotted in Figure 5, but to the granularity +of the graph we cannot tell the two lines apart.} + +\begin{center} +\includegraphics{ps/v105hammfig5.eps}\\ +{\bf Figure 5: Linear Equations Example} +\end{center} + +To be able to decide whether or not a problem is ill-conditioned it is +clearly desirable to have some measure of the condition of a +problem. We show two simple cases where we can obtain such a measure, +and quote the result for a third example. For the first case we derive +the condition number for the evaluation of a function of one variable +[\cite{High02}, Section 1.6]. + +Let $y=f(x)$ with $f$ twice differentiable and $f(x) \ne 0$. +Also let $\hat{y}=f(x+\epsilon)$. Then, using the mean value theorem +\[\begin{array}{rcl} +\hat{y}-y & = & f(x+\epsilon)-f(x)\\ +& = & f^{\prime}(x)\epsilon+ +\displaystyle\frac{f^{\prime\prime}(x+\theta\epsilon)}{2!}\epsilon^2, +~\theta \in (0,1) +\end{array}\] +giving +\[\frac{\hat{y}-y}{y}= +\displaystyle\left(\frac{xf^{\prime}(x)}{f(x)}\right)\frac{\epsilon}{x} ++O(\epsilon^2)\] +The quantity +\[\kappa(x)=\left|\frac{xf^{\prime}(x)}{f(x)}\right|\] + +is called the {\sl condition number} of $f$ since +\[\left|\frac{\hat{y}-y}{y}\right|\approx +\kappa(x)\left|\frac{\epsilon}{x}\right|\] + +Thus if $\kappa(x)$ is large the problem is ill-conditioned, that is small +perturbations in $x$ can induce large perturbations in the solution $y$. + +\subsubsection{Example 4.5} +{\sl Let $y=f(x)=\cos x$. Then we see that} +\[\kappa(x)=|x\tan x|\] +{\sl and, as we might expect, $\cos x$ is most sensitive close to asymptotes +of $\tan x$, such as $x$ close to $\pi/2$.\footnote{The given condition number +is not valid at $x=\pi/2$, since $\cos \pi/2=0$} If we take $x=1.57$ and +$\epsilon=-0.01$ then we find that} +\[\kappa(x)\left|\frac{\epsilon}{x}\right|\approx 12.5577\] +{\sl which is a very good estimate of} $|(\hat{y}-y)/y|=12.55739\ldots$. + +For the second example we consider the condition number of a system of +linear equations $Ax=b$. If we let $\hat{x}$ be the solution of the +perturbed equations +\[(A+E)\hat{x}=b\] +then +\[A(\hat{x}-x)=-E\hat{x},{\rm so\ that\ }\hat{x}-x=-A^{-1}E\hat{x}\] +giving +\[\frac{\Vert\hat{x}-x\Vert}{\Vert\hat{x}\Vert}\le\Vert{}A^{-1}\Vert\cdot +\Vert{}E\Vert=(\Vert A\Vert\cdot\Vert{}A^{-1}\Vert) +\frac{\Vert{}E\Vert}{\Vert{}A\Vert}\eqno{(6)}\] +The quantity +\[\kappa(A)=\Vert{}A\Vert\cdot\Vert{}A^{-1}\Vert\] +is called the condition number of $A$ with respect to the solution of the +equations $Ax=b$, or the condition number of $A$ with respect to matrix +inversion. Since $I=AA^{-1}$, for any norm such that $\Vert{}I\Vert=1$, +we have that $1 \le \kappa(A)$, with equality possible for the $1,2$ and +$\infty$ norms. If A is singular then $\kappa(A)=\infty$. + +\subsubsection{Example 4.6 (Condition of matrix)} +{\sl For the matrix of Example 4.4 we have that} +\[A=\left(\begin{array}{cc} +99 & 98\\ +100 & 99 +\end{array}\right),~\Vert{}A\Vert_1=199\] +{\sl and} +\[A^{-1}=\left(\begin{array}{cc} +99 & -98\\ +-100 & 99 +\end{array}\right),~\Vert{}A^{-1}\Vert_1=199\] +{\sl so that} +\[\kappa_1(A)=199^2\approx 4\times 10^4\] +{\sl Thus we can see that if $A$ is only accurate to about 4 figures, +we cannot guarantee any accuracy in the solution.} + +The term condition number was first introduced by Turing in the context +of systems of linear equations \cite{Turi48}. Note that for an orthogonal +or unitary matrix $Q$, $\kappa(Q)=1$. + +As a third illustration we quote results for the sensitivity of the root +of a polynomial. Consider +\[f(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0\] +and let $\alpha$ be a single root of $f(x)$ so that $f(\alpha)=0$, but +$f^{\prime}(\alpha)\ne 0$. Let $p(x)$ be the perturbed polynomial +\[p(x)=f(x)=\epsilon g(x), +~g(x)=b_nx^n+b_{n-1}x^{n-1}+\ldots+b_1x+b_0\] +with root $\hat{\alpha}=\alpha+\delta$, so that $p(\hat{\alpha})=0$. Then +[\cite{Wilk63},Section 7,Chapter 2] shows that +\[|\delta|\approx +\left|\frac{\epsilon g(\alpha)}{f^{\prime}(\alpha)}\right|\] +Wilkinson also shows that if $\alpha$ is a double root then +\[|\delta|\approx +\left|\left(-\frac{2\epsilon g(\alpha)}{f^{\prime\prime}(\alpha)} +\right)^\frac{1}{2}\right|\] + +\subsubsection{Example 4.7 (Condition of roots of cubic equation)} +{\sl For the root $\alpha=x_1=1$ of the cubic equation of Example 4.1, +with $g(x)=x^3$ and $\epsilon=-0.01$, we have} +\[f^{\prime}(x)=3x^2-42x+120\] +{\sl so that} +\[|\delta|\approx\left|\frac{-0.01\times 1^3}{81}\right|\approx 0.0001\] +{\sl and hence this root is very well-conditioned with respect to +perturbations in the coefficients of $x^3$. On the other hand, for the +double root $\alpha=10$, we have} +\[f^{\prime\prime}(x)=6x-42\] +{\sl so that} +\[|\delta|\approx\left|\left(\frac{-2\times -0.01\times 10^3}{18} +\right)^\frac{1}{2}\right|\approx 1.054\] +{\sl and this time the perturbation of $\epsilon$ produces a rather larger +perturbation in the root. Because $\epsilon$ is not particularly small the +estimate of $\delta$ is not particularly accurate, but we do get a good +warning of the ill-conditioning.} + +Highman [\cite{High02},Section 25.4] +gives a result for the sensitivity of a root of +a general nonlinear equation. + +Problems can be ill-conditioned simply because they are poorly scaled, +often as the result of a poor choice of measurement units. Some algorithms, +or implementation of algorithms, are insensitive to scaling or attempt +automatic scaling, but in other cases a good choice of scaling can be +important to the success of an algorithm. It is also all too easy to turn +a badly scaled problem into a genuinely ill-conditioned problem. + +\subsubsection{Example 4.8 (Badly scaled matrix)} + +{\sl If we let $A$ be the matrix} +\[A=\left(\begin{array}{cc} +2\times 10^9 & 10^9\\ +10^{-9} & 2\times 10^{-9} +\end{array}\right)\] +{\sl then $\kappa_2(A)\approx 1.67\times 10^{18}$ and so $A$ is +ill-conditioned. However we can row scale $A$ as} +\[B=DA= +\left(\begin{array}{cc} +10^{-9} & 0\\ +0 & 10^9 +\end{array}\right) +\left(\begin{array}{cc} +2\times 10^9 & 10^9\\ +10^{-9} & 2\times 10^{-9} +\end{array}\right)= +\left(\begin{array}{cc} +2 & 1\\ +1 & 2 +\end{array}\right)\] +{\sl for which $\kappa_2(B)=3$, so that $B$ is well-conditioned. On the +other hand if we perform a plane rotation on $A$ with $c=0.8$, $s=0.6$ we get} +\[\begin{array}{rcl} +C=GA &=& +\left(\begin{array}{cc} +0.8 & 0.6\\ +-0.6 & 0.8 +\end{array}\right) +\left(\begin{array}{cc} +2\times 10^9 & 10^9\\ +10^{-9} & 2\times 10^{-9} +\end{array}\right)\\ +&&\\ +&=& 2\left(\begin{array}{cc} +8\times 10^8+3\times 10^{-10} & 4\times 10^8+6\times 10^{-10}\\ +-6\times 10^8+4\times 10^{-10} & -3\times 10^8+8\times 10^{-10} +\end{array}\right) +\end{array}\] + +{\sl Since $G$ is orthogonal, +$\kappa_2(C)=\kappa_2(A)\approx 1.67\times 10^{18}$, and so $C$ is of course +as ill-conditioned as $A$, but now scaling cannot recover the situation. +To see that $C$ is genuinely ill-conditioned, we note that} +\[C\approx 2\times 10^8\left(\begin{array}{cc} +8 & 4\\ +-6 & -3 +\end{array}\right)\] +{\sl which is singular. In double IEEE arithmetic, this would be the +floating point represention of $C$.} + +Many of the LAPACK routines perform scaling, or have options to +equilibrate the matrix in the case of linear equations +[\cite{Ande99}, Sections 2.4.1 and 4.4.1], +[\cite{High02}, Sections 7.3 and 9.8], or to balance in the case of +eigenvalue problems [\cite{Ande99}, Sections 4.8.1.2 and 4.11.1.2]. + +\subsection{Stability} + +The {\sl stability} of a method for solving a problem is concerned with the +sensitivity of the method to (rounding) errors in the solution +process. A method that guarantees as accurate a solution as the data +warrants is said to be stable, otherwise the method is unstable. To +emphasise the point we note that, whereas condition is concerned with +the sensitivity of the problem, stability is concerned with the +sensitivity of the method of solution. + +An example of an unstable method is that of (3) for computing sample +variance. We now give two more simple illustrative examples. + +\subsubsection{Example 4.9 (Quadratic equation)} + +{\sl Consider the quadratic equation} +\[1.6x^2-100.1x+1.251=0\] +{\sl Four significant figure arithmetic when using the standard formula} +\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\] +{\sl gives} +\[x_1=62.53,~x_2=0.03125\] +{\sl If we use the relationship $x_1x_2=c/a$ to compute $x_2$ from $x_1$ +we instead find that} +\[x_2=0.01251\] +{\sl The correct solution is $x_1=62.55$,$x_2=0.0125$. We can see that in +using the standard formula to compute the smaller root we have suffered +from cancellation, since $\sqrt{b^2-4ac}$ is close to $(-b)$.} + +Even a simple problem such as computing the roots of a quadratic equation +needs great care. A very nice discussion is to be found in \cite{Fors69}. + +\subsubsection{Example 4.10 (Recurrence relation)} + +{\sl Consider the computation of $y_n$ defined by} +\[y_n=(1/e)\int_0^1{x^ne^x~dx}\eqno{(7)}\] +{\sl where $n$ is a non-negative integer. We note that, since in the interval +$[0,1]$, $(1/e)e^x$ is bounded by unity, it is easy to show that} +\[0\le y_n\le 1/(n+1)\eqno{(8)}\] + +{\sl Integrating (7) by parts gives} +\[y_n=1-ny_{n-1},~y_0=1-1/e=0.6321205882856\ldots\eqno{(9)}\] +{\sl and we have a seemingly attractive method for computing $y_n$ for a +given value of $n$. The result of using this forward recurrence +relation, with IEEE double precision arithmetic, to compute the values +of $y_i$ up to $y_21$ is shown in Table 2. Bearing in mind the bounds of +(8), we see that later values are diverging seriously from the correct +solution.} + +{\sl A simple analysis shows the reason for the +instability. Since y 0 cannot be represented exactly, we cannot avoid +starting with a slightly perturbed value, $\hat{y}_0$ . So let} +\[\hat{y}_0=y_0+\epsilon\] + +%\begin{tabular}{|c|c|c|c|c|c|c|c|} +%\hline +%$y_0$ & $y_1$ & $y_2$ & $y_3$ & $y_4$ & $y_5$ & $y_6$ & $y_7$\\ +%0.6321 & 0.3679 & 0.2642 & 0.2073 & 0.1709 & 0.1455 & 0.1268 & 0.1124\\ +%\hline +%$y_8$ & $y_9$ & $y_{10}$ & $y_{11}$ & +%$y_{12}$ & $y_{13}$ & $y_{14}$ & $y_{15}$\\ +%0.1009 & 0.0916 & 0.0839 & 0.0774 & 0.0718 & 0.0669 & 0.0627 & 0.0590\\ +%\hline +%$y_{16}$ & $y_{17}$ & $y_{18}$ & $y_{19}$ & $y_{20}$ & $y_{21}$\\ +%0.0555 & 0.0572 & -0.0295 & 1.5596 & -30.1924 & 635.0403\\ +%\hline +%\end{tabular} + +\begin{center} +\includegraphics[scale=0.75]{ps/v105hammtable2.eps}\\ +{\bf Table 2: Forward Recurrence for $y_n$} +\end{center} + +{\sl Then, even if the remaining computations are performed exactly we +see that} +\[\begin{array}{ccccc} +\hat{y}_1 &=& 1-\hat{y}_0 &=& y_1-\epsilon\\ +\hat{y}_2 &=& 1-2\hat{y}_1 &=& y_2+2\epsilon\\ +\hat{y}_3 &=& 1-3\hat{y}_2 &=& y_3-6\epsilon\\ +\hat{y}_4 &=& 1-4\hat{y}_3 &=& y_4+24\epsilon +\end{array}\] +{\sl and a straightforward inductive proof shows that} +\[\hat{y}_n=y_n+(-1)^nn!\epsilon\] +{\sl When $n=21$,$n!\approx 5.1091\times 10^{19}$. We see clearly that this +forward recurrence is an unstable method of computing $y_n$, since the +error grows rapidly as we move forward.} + +The next example illustrates a stable method of computing $y_n$. + +\subsubsection{Example 4.11 (Stable recurrence)} + +{\sl Rearranging (9) we obtain the backward recurrence} +\[y_{n-1}=(1-y_n)/n\] +{\sl Suppose that we have an approximation, $\hat{y}_{n+m}$, +to $y_{n+m}$ and we let} +\[\hat{y}_{n+m}=y_{n+m}+\epsilon\] +{\sl Then, similarly to the result of Example 4.10, we find that} +\[\hat{y}_n=y_n+ +\frac{(-1)^{m-n}\epsilon}{(n+m)(n+m-1)\ldots(n+1)}\] +{\sl and this time the initial error decays rapidly, rather than grows +rapidly as in Example 4.10. If we take an initial guess of $y_{21}=0$, +we see from (8) that} +\[|\epsilon| \le 1/21 \le 0.05\] +{\sl Using this backward recurrence relation, with IEEE double precision +arithmetic, gives the value} +\[y_0=0.63212055882856\] +{\sl which is correct to all the figures shown. We see that this +backward recurrence is stable.} + +It should also be said that the integral of (7) can be evaluated +stably without difficulty using a good numerical integration +(quadrature) formula, since the function $f(x) = (1/e)x^n e^x$ is +non-negative and monotonic throughout the interval [0,1]. + +In the solution of ordinary and partial differential equations one +form of instability can arise by replacing a differential equation by +a difference equation. We first illustrate the problem by the solution +of a simple nonlinear equation. + +\subsubsection{Example 4.12 (Parasitic solution)} + +{\sl The equation} +\[e^{-x}=99x\eqno{(10)}\] +{\sl has a solution close to $x=0.01$. By expanding $e^{-x}$ as a power +series we have that} +\[e^{-x}=1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\ldots\approx +1-x+\frac{x^2}{2!}\] +{\sl and hence an approximate solution of (10) is a root of the +quadratic equation} +\[x^2-200x+2=0\] +{\sl which has the two roots $x_1\approx 0.0100005$, $x_2\approx 199.99$. +The second root clearly has nothing to do with the original equation and +is called a {\bf parasitic} solution.} + +In the above example we are unlikely to be fooled by the parasitic solution, +since it so clearly does not come close to satisfying (10). But in the +solution of ordinary or partial differential equations such bifurcurations, +due to truncation error, may not be so obvious. + +\subsubsection{Example 4.13 (Instability for ODE)} + +{\sl For the initial value problem} +\[y^{\prime}=f(x,y),~y=y_0{\rm\ when\ }x=x_0\eqno{(11)}\] +{\sl the mid-point rule, or leap-frog method, for solving the differential +equation is given by} +\[y_{r+1}=y_{r-1}+2hf_r\eqno{(12)}\] +{\sl where $h=x_i-x_{i-1}$ for all $i$ and $f_r=f(x_r,y_r)$. This method +has a truncation error of $O(h^3)$ [\cite{Isaa94}, Section 1.3, +Chapter 8]\footnote{Here it is called the centered method. It is an +example of a Nystr\"om method.} This method requires two starting values, +so one starting value must be estimated by some other method. Consider +the case where} +\[f(x,y)=\alpha y,~y_0=1,~x_0=0\] +{\sl so that the solution of (11) is $y=e^{\alpha x}$. Figures 6 and 7 +show the solution obtained by using (12) when $h=0.1$ for the cases +$\alpha=2.5$ and $\alpha=-2.5$ respectively. In each case the value of +$y_1$ is taken as the correct four figure value, $y_1=1.284$ when +$\alpha=2.5$ and $y_1=0.7788$ when $\alpha=-2.5$. We see that in the +first case the numerical solution does a good job in following the +exact solution, but in the second case oscillation sets in and the +numerical solution diverges from the exact solution.} + +The reason for this behaviour in the above example is that (12) has +the solution +\[y_r=A\left(\alpha h+(1+\alpha^2h^2)^\frac{1}{2}\right)^r+ +B\left(\alpha h-(1+\alpha^2h^2)^\frac{1}{2}\right)^r\eqno{(13)}\] + +where A and B are constants that depend on the intial conditions. With +the initial conditions $y_0 = 1$, $x_0 = 0$ and $y_1 = e^{αh}$, $x_1 = h$ +we find that $A = 1+O(h^3)$, $B=O(h^3)$. We can see that the first term +in (13) approximates the exact solution, but the second term is a +parasitic solution. When $\alpha > 0$ the exact solution increases and the +parasitic solution decays, and so is harmless, but when $\alpha < 0$ the +exact solution decays and the parasitic solution grows as illustrated +in Figure 7. An entertaining discussion, in the context of the +Milne-Simpson method, of the above phenomenon is given in Acton +[\cite{Acto70}, Chapter 5], a book full of good advice and insight. A more +recent book by Action in the same vein is \cite{Acto96}. + +\begin{center} +\includegraphics{ps/v105hammfig6.eps}\\ +{\bf Figure 6: Stable ODE Example} +\end{center} + +\begin{center} +\includegraphics{ps/v105hammfig7.eps}\\ +{\bf Figure 7: Unstable ODE Example} +\end{center} + +\subsection{Error Analysis} + +{\sl Error analysis} is concerned with analysing the cumulative effects +of errors. Usually these errors will be rounding or truncation errors. +For example, if the polynomial +\[p(x)=p_0+p_1x+p_2x^2+\cdots+p_nx^n\] +is evaluated at some point $x=\alpha$ using Horner's scheme (nested +multiplication) as +\[p(\alpha)=p_0+\alpha(p_1+\cdots+\alpha(p_{n-2}+ +\alpha(p_{n-1}+\alpha{}p_n))\ldots)\] + +we might ask under what conditions, if any, on the coefficients +$p_0,p_1,\ldots,p_n$ and $\alpha$, the solution will, in some sense, be +reasonable? To answer the question we need to perform an error +analysis. + +Error analysis is concerned with establishing whether or +not an algorithm is stable for the problem in hand. A {\sl forward error +analysis} is concerned with how close the computed solution is to the +exact solution. A {\sl backward error analysis} is concerned with how well +the computed solution satisfies the problem to be solved. On first +acquaintance, that a backward error analysis, as opposed to a forward +error analysis, should be of interest often comes as a surprise. The +next example illustrates the distinction between backward and forward +errors. + +\subsubsection{Example 4.14 (Linear equations)} +{\sl Let} +\[A=\left(\begin{array}{cc} +99 & 98\\ +100 & 99 +\end{array}\right){\rm\ and\ } +b=\left(\begin{array}{c} +1\\ +1 +\end{array}\right)\] +{\sl Then the exact solution of the equations $Ax=b$ is given by} +\[x=\left(\begin{array}{c} +1\\ +-1 +\end{array}\right)\] +{\sl Also let $\hat{x}$ be an approximate solution to the equations and +define the {\bf residual} vector $r$ as} +\[r=b-A\hat{x}\eqno{(14)}\] +{\sl Of course, for the exact solution $r=0$ and we might hope that for +a solution close to $x$, $r$ should be small. Consider the approximate +solution} +\[\hat{x}=\left(\begin{array}{c} +2.97\\ +-2.99 +\end{array}\right),{\rm\ for\ which\ }\hat{x}-x= +\left(\begin{array}{c} +1.97\\ +-1.99 +\end{array}\right)\] +{\sl and so $\hat{x}$ looks to be a rather poor solution. But for this +solution we have that} +\[r=\left(\begin{array}{c} +-0.01\\ +0.01 +\end{array}\right)\] +{\sl and we have almost solved the original problem. On the other hand +the approximate solution} +\[\hat{x}=\left(\begin{array}{c} +1.01\\ +-0.99 +\end{array}\right),{\rm\ for\ which\ }\hat{x}-x= +\left(\begin{array}{c} +0.01\\ +0.01 +\end{array}\right)\] +{\sl gives} +\[r=\left(\begin{array}{c} +-1.97\\ +-1.97 +\end{array}\right)\] +{\sl and, although $\hat{x}$ is close to $x$, it does not solve a problem +close to the original problem.} + +Once we have computed the solution to a system of linear equations +$Ax=b$ we can, of course, readily compute the residual of (14). If we +can find a matrix $E$ such that +\[E\hat{x}=r\eqno{(15)}\] +then +\[(A+E)\hat{x}=b\] +and we thus have a measure of the perturbation in $A$ required to make +$\hat{x}$ an exact solution. A particular $E$ that satisfies (15) is +given by +\[E=\frac{r\hat{x}^T}{\hat{x}^T\hat{x}}\] + +From this equation we have that +\[\Vert E\Vert_2 \le +\frac{\Vert r\Vert_2\Vert\hat{x}\Vert_2}{\Vert\hat{x}\Vert_2^2} +=\frac{\Vert r\Vert_2}{\Vert\hat{x}\Vert_2}\] + +Thus, this particular $E$ minimizes $\Vert E\Vert_2$. Since +$x^Tx=\Vert x\Vert_F^2$ it also minimizes $E$ in the Frobenious norm. +This gives us an {\sl a posteriori} bound on the backward error. + +\subsubsection{Example 4.15 (Perturbation in linear equations)} + +{\sl Consider the equations $Ax=b$ of Example 4.14 and the 'computed' +solution} +\[\hat{x}=\left(\begin{array}{c} +2.97\\ +-2.99 +\end{array}\right){\rm\ for\ which\ } +r=\left(\begin{array}{c} +-0.01\\ +0.01 +\end{array}\right)\] +{\sl Then} +\[r\hat{x}^T=\left(\begin{array}{cc} +-0.0297 & 0.0299\\ +0.0297 & -0.0299 +\end{array}\right),~\hat{x}^T\hat{x}=17.761\] +{\sl and} +\[E\approx +\left(\begin{array}{cc} +-0.00167 & 0.00168\\ +0.00167 & -0.00168 +\end{array}\right)\] + +{\sl Note that $\Vert E\Vert_F/\Vert A\Vert_F\approx 1.695\times 10^{-5}$ +and so the computed solution corresponds to a small relative perturbation +in $A$.} + +From (6) we have that +\[\frac{\Vert\hat{x}-x\Vert}{\Vert\hat{x}\Vert} \le +\kappa(A)\frac{\Vert E\Vert}{\Vert A\Vert}\] +and so, if we know $\kappa(A)$, then an estimate of the backward error +allows us to estimate the forward error. + +As a general rule, we can say that approximately: +\[{\rm forward\ error\ } \le {\rm\ condition\ error\ }\times +{\rm\ backward\ error}\] + +Although the idea of backward error analysis had been introduced by +others, it was James Hardy Wilkinson who really developed the theory +and application, and gave us our understanding of error analysis and +stability, particularly in the context of numerical linear +algebra. See the classic books \cite{Wilk63} and \cite{Wilk65}. + +A wonderful modern book that continues +the Wilkinson tradition is \cite{High02}. The solid foundation for the +numerical linear algebra of today relies heavily on the pioneering +work of Wilkinson; see also \cite{Wilk71}\footnote{ +In Givens 1954 technical report quoted earlier \cite{Give54}, which +was never published in full and must be one of the most oft quoted +technical reports in numerical analysis, as well as the introduction +of Givens plane rotations, it describes the use of Sturm sequences for +computing eigenvalues of tridiagonal matrices, and contains probably +the first explicit backward error analysis. Wilkinson, who so successfully +developed and expounded the theory and analysis of rounding errors, +regarded the {\sl a priori} error analysis of Givens as ``one of the +landmarks in the history of the subject'' [\cite{Wilk65}, Additional +notes to Chapter 5]} + +Wilkinson recognised that error analysis could be tedious and often +required great care, but was nevertheless essential to our +understanding of the stability of algorithms. + +\begin{quote} + +``The clear identification of the factors determining the stability of +an algorithm soon led to the development of better algorithms. The +proper understanding of inverse iteration for eigenvectors and the +development of the QR algorithm by Francis are the crowning +achievements of this line of research. + +For me, then, the primary purpose of the rounding error analysis was insight.” +\cite{Wilk86}, p. 197. +\end{quote} + +As a second example to illustrate forward and backward errors, +we consider the quadratic equation of Example 4.9. + +\subsubsection{Example 4.16 (Errors in quadratic equation)} + +{\sl For the quadratic equation of Example 4.9 we saw that the +standard formula gave the roots $x_1=62.53$, $x_2=0.03125$. Since the +correct solution is $x_1=62.55$, $x_2=0.0125$ the second root has a +large forward error. If we form the quadratic +$q(x)=1.6(x−x_1)(x−x_2)$, +rounding the answer to four significant figures, we get} +\[q(x)=1.6x^2-100.1x+3.127\] +{\sl and the constant term differs significantly from the original value of +1.251, so that there is also a large backward error. The standard +method is neither forward nor backward stable. On the other hand, for +the computed roots $x_1=62.53$, $x_2=0.01251$ we get} +\[q(x)=1.6x^2-100.1x+1.252\] +{\sl so this time we have both forward and backward stability.} + +An example of a computation that is forward stable, but not backward +stable is that of computing the coefficients of the polynomial +\[p(x)=(x-x_1)(x-x_2)\cdots(x-x_n),~x_i>0\] + +In this case, since the $x_i$ are all of the same sign, no cancellation +occurs in computing the coefficients of $p(x)$ and the computed +coefficients will be close to the exact coefficients; thus we have +small forward errors. On the other hand, as Example 4.1 illustrates, +the roots of polynomials can be sensitive to perturbations in the +coefficients and so the roots of the computed polynomial could differ +significantly from $x_1,x_2,\ldots,x_n$. + +\subsubsection{Example 4.17 (Ill-conditioned polynomial)} + +{\sl The polynomial whose roots are $x_i=i$, $i=1,2,\ldots,20$, is} +\[p(x)=x^{20}-210x^{19}+\ldots+20!\] + +{\sl Suppose that the coefficient of $x^{19}$ is computed as +$-(210+2^{-23})$; then we find that $x^{16}$, $x^{17}\approx 16.73\pm2.813i$. +Thus a small error in computing a coefficient produced a polynomial with +significantly different roots from those of the exact polynomial. This +polynomial is discussed in Wilkinson [\cite{Wilk63}, Chapter 2, Section 9] and +\cite{Wilk84}. See also Wilkinson [\cite{Wilk85}, Section 2].} + +\section{Floating Point Error Analysis} + +{\sl Floating point error analysis} is concerned with the analysis of +errors in the presence of floating point arithmetic. It is based on the +relative errors that result from each basic operation. We give just a +brief introduction to floating point error analysis in order to illustrate +the ideas. + +Let $x$ be a real number; then we use the notation fl($x$) to represent +the floating point value of $x$. The fundamental assumption is that +\[fl(x)=x(1+\epsilon),~|\epsilon| \le u\eqno(16)\] +where $u$ is the unit roundoff of (1). Of course, +\[\frac{fl(x)-x}{x}=\epsilon\] + +A useful alternative is +\[fl(x)=\frac{x}{1+\delta},~|\delta| \le u,~{\rm so\ that\ } +\frac{fl(x)-x}{fl(x)}=\delta\eqno{(17)}\] + +\subsection{Example 5.1 (Floating point numbers)} +{\sl Consider four figure decimal arithmetic with} +\[u=\frac{1}{2}\times 10^{-3}=5\times 10^{-4}\] + +{\sl If $x=\sqrt{2}=1.414213\ldots$ then $fl(x)=1.414$ and} +\[|\epsilon|=\left|\frac{fl(x)-x}{x}\right|\approx 1.5\times 10^{-4}\] + +{\sl If $x=1.000499\ldots$ then $fl(x)=1.000$ and} +\[|\epsilon|=\left|\frac{fl(x)-x}{x}\right|\approx 5\times 10^{-4}=u\] + +{\sl If $x=1000.4999\ldots$ then fl(x)=1000 and again} +\[|\epsilon|=\left|\frac{fl(x)-x}{x}\right|\approx 5\times 10^{-4}=u\] + +Bearing in mind (16), if $x$ and $y$ are floating point numbers, then the +standard model of floating point arithmetic, introduced by Wilkinson +\cite{Wilk60}, is given by +\[fl(x\otimes y)=(x\otimes y)(1+\epsilon),~|\epsilon|\le u\] +\[{\rm where\ }\otimes \equiv +,-,\times,\div\] + +It is assumed, of course, that $x\otimes y$ produces a value that is in the +range of representable floating point numbers. Comparable to (17), a useful +alternative is +\[fl(x\otimes y)=\frac{x\otimes y}{1+\delta},~|\delta|\le u\] + +When we consider a sequence of floating point operations we frequently +obtain products of error terms of the form +\[(1+\epsilon)=(1+\epsilon_1)(1+\epsilon_2)\ldots(1+\epsilon_r)\] +so that +\[(1-u)^r\le1+\epsilon\le(1+u)^r\] + +If we ignore second order terms then we have the reasonable assumption +that\footnote{Those who are uncomfortable with the approximation may +prefer to replace the bound $|\epsilon|\le ru$ with one of the form +$|\epsilon|\le \gamma_r$, where $\gamma_r=(ru)/(1-ru)$ and $ru < 1$ +is assumed. See \cite{High02}, Lemma 3.1} +\[|\epsilon| \le ru\eqno{(19)}\] + +We now give three illustrative examples. In all three examples the $x_i$ +are assumed to be floating point numbers, that is, they are values +that are already represented in the computer. This is, of course, a +natural assumption to make when we are analysing the errors in a +computation. + +\subsection{Example 5.2 (Product of values)} +{\sl Let $x=x_0x_1\ldots x_n$ and $\tilde{x}=fl(x)$. Thus we have $n$ +products to form, each one introducing an error bounded by $u$. Hence +from (18) we get} +\[\tilde{x}=x_0x_1(1+\epsilon_1)x_2(1+\epsilon_2)\ldots +x_n(1+\epsilon_n),~|\epsilon_i|\le u\eqno{(20)}\] +{\sl and from (19) we see that} +\[\tilde{x}=x(1+\epsilon),~|\epsilon|\le nu\eqno{(21)}\] +{\sl where} +\[1+\epsilon=(1+\epsilon_1)(1+\epsilon_2)\ldots(1+\epsilon_n)\] + +{\sl We can see from (21) that this computation is forward stable, because +the result is close to the exact result, and from (20) the computation is +also backward stable, because the result is exact for a slightly perturbed +problem; that is the result is exact for the data} +$x_0$,$x_1(1+\epsilon_1)$,$x_2(1+\epsilon_2)$,\ldots,$x_n(1+\epsilon_n)$. + +\subsection{Example 5.3 (Sum of values)} + +{\sl Let $s=x_1+x_2+\ldots+x_n$ and $\tilde{s}=fl(s)$. By considering} +\[s_r=fl(s_{r-1}+x_r),~s_1=x_1\] +{\sl it is straightforward to show that} +\[\begin{array}{ccc} +\tilde{s}&=&x_1(1+\epsilon_1)+x_2(1+\epsilon_1)+x_3(1+\epsilon_2)+\ldots+ +x_n(1+\epsilon{n-1})\\ +&=&x+(x_1\epsilon_1+x_2\epsilon_1+x_3\epsilon_2+\ldots+ +x_n\epsilon_{n-1}),~|\epsilon_r|\le(n-r+1)u +\end{array}\] + +{\sl Here we see that summation is backward stable, but is not necessarily +forward stable. Example 3.1 gives a case where summation is not +forward stable, but notice that the computed solution is the exact +solution of the slightly perturbed problem} +\[1.000+1.000\times 10^4-1.0001\times 10^4=0\] +{\sl which illustrates the backward stability.} + +Note that if the $x_i$ all have the same sign, then summation is forward +stable because +\[|\tilde{s}-s|\le(|x_1|+|x_2|+\ldots+|x_n|)nu=|s|nu\] +so that +\[\frac{|\tilde{s}-s|}{|s|}\le nu,~s\ne 0\] + +\subsection{Example 5.4 (Difference of two squares)} + +{\sl Consider the computation} +\[z=x^2-y^2\eqno{(22)}\] + +{\sl We can, of course, also express $z$ as} +\[z=(x+y)(x-y)\eqno{(23)}\] + +{\sl If we compute $z$ from (22) we find that} +\[\begin{array}{rcl} +\tilde{z}&=&fl(z^2-y^2)=x^2(1+\epsilon_1)=y^2(1=\epsilon_2)\\ +&=&z+(x^2\epsilon_1-y^2\epsilon_2),~\epsilon_1,\epsilon_2\le 2u +\end{array}\] +{\sl and so this is backward stable, but not forward stable. +On the other hand, if we compute $z$ from (23)} +\[\begin{array}{rcl} +\hat{z}&=&fl((x+y)(x-y))=(x+y)(x-y)(1+\epsilon)\\ +&=&z(1+\epsilon),~\epsilon \le 3u +\end{array}\] +{\sl and so this is both backward and forward stable. As an +example, if we take} +\[x=543.2,~y=543.1,~{\rm\ so\ that\ }z=108.63\] +{\sl and use four significant figure arithmetic we find that} +\[\tilde{z}=100,~but \hat{z}=108.6\] +{\sl Clearly $\tilde{z}$ has suffered from cancallation, but $\hat{z}$ +has not.} + +We now quote some results, without proof, of solving higher level +linear algebra problems to illustrate the sort of results that are +possible. Principally we consider the solution of the n linear +equations +\[Ax=b\eqno{(24)}\] + +by Gaussian elimination and we assume that the reader is familiar with +Gaussian elimination. The $k$th step of Gaussian elimination can be +expressed as +\[A_k=M_kP_kA_{k-1}Q_k,~A_0=A\eqno{(25)}\] + +where $P_k$ and $Q_k$ are permutation matrices, one or both of which may +be the unit matrix, chosen to implement whatever pivoting strategy is +used and $M_k$ is the multiplier matrix chosen to eliminate the elements +below the diagonal of the $k$th column of $A_{k−1}$. This results in the +factorization +\[A=PLUQ\] + +where $P$ and $Q$ are permutation matrices, $L$ is a unit lower triangular +matrix and $U$ is upper triangular. To simplify analysis it is usual +to assume that, with hindsight, $A$ has already been permuted so that we +can work with $A\Leftarrow P^T AQ^T$. In this case (25) becomes +\[A_k=M_kA_{k-1},~A_0=A\] + +and $M_k$ and $A_{k-1}$ have the form +\[M_k= +\left(\begin{array}{ccc} +I & 0 & 0\\ +0 & 1 & 0\\ +0 & -m_k & I +\end{array}\right),~A_{k-1}= +\left(\begin{array}{ccc} +U_{k-1} & u_{k-1} & X_{k-1}\\ +0 & \alpha_{k-1} & b_{k-1}^T\\ +0 & a_{k-1} & \hat{A}_{k-1} +\end{array}\right)\] +$m_k$ is chosen to eliminate $a_{k-1}$, so that +\[a_{k-1}-\alpha_{k-1}m_k=0,~{\rm\ giving\ }m_k=a_{k-1}/\alpha_{k-1}\] +$\hat{A}_{k-1}$ is updated as +\[\tilde{A}_{k}=\hat{A}_{k-1}-m_kb_{k-1}^T\equiv +\left(\begin{array}{cc} +\alpha_k & b_k^T\\ +a_k & \hat{A}_k +\end{array}\right)\] +and +\[A=LU,~{\rm\ where\ }L=M_1^{-1}M_2^{-1}\ldots M_{n-1}^{-1},~{\rm\ and\ } +U=A_{n-1}\] +Since +\[M_k^{-1}= +\left(\begin{array}{ccc} +I & 0 & 0\\ +0 & 1 & 0\\ +0 & m_k & I +\end{array}\right)\] +we have that +\[L= +\left(\begin{array}{ccccc} +1 & 0 & \cdots & 0 & 0\\ +m_{21} & 1 & \cdots & 0 & 0\\ +m_{31} & m_{32} & \cdots & 0 & 0\\ +\vdots & \vdots && \vdots & \vdots\\ +m_{n-1,1} & m_{n-1,2} & \cdots & 1 & 0\\ +m_{n1} & m_{n2} & \cdots & m_{n,n-1} & 1 +\end{array}\right)\] + +It can be shown that the computed factors $\tilde{L}$ and $\tilde{U}$ satisfy +\[\tilde{L}\tilde{U}=A+F\] +where various bounds on $F$ are possible; for example, for the 1, +$\infty$ or $F$ norms +\[\Vert F\Vert\le 3ngu\Vert A\Vert, +~g=\frac{{\rm max}\Vert\tilde{A}_k\Vert}{\Vert A\Vert}\] +$g$ is called the {\sl growth factor}. Similarly it can be shown that the +computed solution of (24), $\tilde{x}$, satisfies +\[(A+E)\hat{x}=b\] +where a typical bound is +\[\Vert E\Vert\le 3n^2gu\Vert A\Vert\] + +We can see that this bound is satisfactory unless $g$ is large, so it is +important to choose $P$ or $Q$, or both, in order to control the size of +$g$. This is essentially the classic result of Wilkinson\cite{Wilk61} and +Wilkinson [\cite{Wilk63}, Section 25], +where the $\infty$-norm is used and the use +of partial pivoting is assumed; see also Higham [\cite{High02}, Theorem 9.5]. + +The next example gives a simple demonstration of the need for pivoting. + +\subsection{Example 5.5 (The need for pivoting)} +{\sl Consider the matrix} +\[A=\left(\begin{array}{cc} +0.001 & 12\\ +10 & -10 +\end{array}\right)\] +{\sl and the use of four significant figure arithmetic. Since this is just a +two by two matrix we have that $M_1^{-1}=L$ and $M_1A=U$. Denoting the +computed matrix $X$ by $\tilde{X}$, we find that} +\[L=\tilde{L}=\left(\begin{array}{cc} +1 & 0 \\ +10000 & 1 +\end{array}\right),~U= +\left(\begin{array}{cc} +0.001 & 12\\ +0 & -120010 +\end{array}\right)~and~\tilde{U}= +\left(\begin{array}{cc} +0.001 & 12\\ +0 & -120000 +\end{array}\right)\] +{\sl which gives} +\[U=\tilde{U}=\left(\begin{array}{cc} +0 & 0\\ +0 & 10 +\end{array}\right)\] +{\sl and} +\[F=\tilde{L}\tilde{U}=A-\left(\begin{array}{cc} +0 & 0\\ +0 & 10 +\end{array}\right)=U-\tilde{U}\] + +{\sl Thus whilst $\Vert F\Vert$ is small relative to $\Vert U\Vert$, +it corresponds to a large relative perturbation in $\Vert A\Vert$. +On the other hand if we permute the two rows of $A$ to give} +\[\overline{A}=\left(\begin{array}{cc} +10 & -10\\ +0.001 & 12 +\end{array}\right)\] +{\sl we have that} +\[L=\tilde{L}=\left(\begin{array}{cc} +1 & 0\\ +0.0001 & 1 +\end{array}\right),~U=\left(\begin{array}{cc} +10 & -10\\ +0 & 12.001 +\end{array}\right){\rm\ and\ } +\tilde{U}=\left(\begin{array}{cc} +10 & -10\\ +0 & 12.00 +\end{array}\right)\] +{\sl which gives} +\[U-\tilde{U}=\left(\begin{array}{cc} +0 & 0\\ +0 & -0.001 +\end{array}\right)\] +{\sl and} +\[F=\tilde{L}\tilde{U}-A=\left(\begin{array}{cc} +0 & 0\\ +0 & -0.001 +\end{array}\right)=U-\tilde{U}\] +{\sl This time $\Vert F\Vert$ is small relative to both +$\Vert U\Vert$ and $\Vert A\Vert$.} + +If we put $m={\rm max}|\tilde{m}_{ij}|$ then we can show that +\[g\le (1+m)^{n-1}\] +Partial pivoting ensures that +\[m\le 1{\rm\ and\ hence\ }g\le 2^{n-1}\] +Only very special examples get anywhere near this bound, one example due to +Wilkinson being matrices of the form +\[A=\left(\begin{array}{cccccc} +1 & 0 & 0 & \cdots & 0 & 1\\ +-1 & 1 & 0 & \cdots & 0 & 1\\ +-1 & -1 & 1 & \cdots & 0 & 1\\ +\vdots & \vdots & \vdots & & \vdots & \vdots\\ +-1 & -1 & -1 & \cdots & 1 & 1\\ +-1 & -1 & -1 & \cdots & -1 & 1 +\end{array}\right) +~{\rm\ for\ which\ }U= +\left(\begin{array}{cccccc} +1 & 0 & 0 & \cdots & 0 & 1\\ +0 & 1 & 0 & \cdots & 0 & 2\\ +0 & 0 & 1 & \cdots & 0 & 4\\ +\vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ +0 & 0 & 0 & \cdots & 1 & 2^{n-2}\\ +0 & 0 & 0 & \cdots & 0 & 2^{n-1} +\end{array}\right)\] + +Despite such examples, in practice partial pivoting is the method of +choice, but careful software should at least include an option to +monitor the growth factor. + +There are classes of matrices for which pivoting is not needed to +control the growth of $g$ [\cite{High02}, Table 9.1]. Perhaps the most +important case is that of symmetric positive definite matrices for +which it is known a priori that growth cannot occur, and so Gaussian +elimination is stable when applied to a system of equations for which +the matrix of coefficients is symmetric positive definite\footnote{ +The variant of Gaussian elimination that is usually used in this case +is {\sl Cholesky's method}} + +The choice of pivots is affected by scaling and equilibration, and a +poor choice of scaling can lead to a poor choice of pivots. A full +discussion on pivoting strategies, equilibration and scaling, as well +as sage advice, can be found in \cite{High02}. + +For methods that use orthogonal transformations we can usually obtain +similar error bounds, but without the growth factor, since orthogonal +transformations preserve the 2−norm and $F$−norm. For example, if we +use Householder transformations to perform a $QR$ factorization of $A$ for +the solution of the least squares problem +min$_x\Vert b-Ax\Vert_2$ , where $A$ is +an $m$ by $n$, $m \ge n$ matrix of rank $n$ \cite{Golu65}, +the computed solution $\tilde{x}$ satisfies +\[\underset{x}{\rm min}\Vert(b+f)-(A+E)\tilde{x}\Vert_2\] +where $f$ and $E$ satisfy bounds of the form +\[\Vert f\Vert_F\le c_1mnu\Vert b\Vert_F, +~\Vert E\Vert_F\le c_2mnu\Vert A\Vert_F\] + +and $c_1$ and $c_2$ are small integer constants [\cite{Laws95}, page 90]. + +Similarly, for the solution of the eigenvalue problem $Ax = \lambda x$, +where $A$ is an $n$ by $n$ matrix, using Housholder transformations to +reduce $A$ to upper Hessenberg form, followed by the $QR$ algorithm to +further reduce the Hessenberg form to upper triangular Schur form, the +computed solution satisifies +\[(A+E)\tilde{x}=\tilde{\lambda}\tilde{x}\] +where +\[\Vert E\Vert_F \le p(n)u\Vert A\Vert_F\] +and $p(n)$ is a modestly growing function of $n$ +\cite{Wilk65}; \cite{Ande99}. + +We note that the bounds discussed so far are called {\sl normwise} bounds, +but in many cases they can be replaced by {\sl componentwise} bounds which +bound the absolute values of the individual elements, and so are +rather more satisfactory. For instance, if $A$ is a sparse matrix, we +would probably prefer not to have to perturb the elements that are +structurally zero. As a simple example, consider the triangular +equations +\[Tx=b,~T-n{\rm\ by\ }n{\rm\ triangular}\] + +and let $\tilde{x}$ be the solution computed by forward or backward +substitution, depending on whether $T$ is lower or upper triangular +respectively. Then it can readily be shown that $\tilde{x}$ satisifes +\[(T+E)\tilde{x}=b,~{\rm\ with\ }|e_{ij}|\le nu|t_{ij}|\] + +which is a strong componentwise result showing backward stability +[\cite{High02}, Theorem 8.5]. + +Associated with componentwise error bounds are componentwise condition +numbers. Once again see \cite{High02} for further details and +references. + +\section{Posing the Mathematical Problem} + +In this short section we merely wish to raise awareness of the need to +model a problem correctly, without offering any profound solution. + +It can be all too easy to transform a well-conditioned problem into an +ill-conditioned problem. For instance, in Example 4.10 we transformed +the well-conditioned quadrature problem of finding +\[y_n=(1/e)\int_0^1{x^ne^x~dx},~n\ge 0\] +into the ill-conditioned problem of finding $y_n$ from the forward +recurrence relation +\[y_n=1-ny_{n-1},~y_0=1-1/e\] + +As another example, we noted in Section 4.3 that polynomials can be +very ill-conditioned. It follows that the eigenvalues of a matrix $A$ +should most certainly not be computed via the characteristic equation +of $A$. For example, if $A$ is a symmetric matrix with eigenvalues +$\lambda i=i,~i=1,2,\ldots,20$, then the characteristic equation of $A$, +det($A-\lambda{}A$), is very ill-conditioned (see Example 4.17). + +On the other hand, the eigenvalues of a symmetric matrix are always +well-conditioned [\cite{Wilk65}, Section 31, Chapter 2]. + +The above two examples illustrate the dangers in transforming the +mathematical problem. Sometimes it can be poor modelling of the +physical problem that gives rise to an ill-conditioned mathematical +problem, and so we need to think carefully about the whole modelling +process. + +We cannot blame software for giving us poor solutions if we provide +the wrong problem. We can, of course, hope that the software might +provide a measure for the condition of the problem, or some measure of +the accuracy of the solution to give us warning of a poorly posed +problem. + +At the end of Section 4.1 we also mentioned the desirability of +careful choice of measurement units, in order to help avoid the +effects of poor scaling. + +\section{Error Bounds and Software} + +In this section we give examples of reliable software that return +information about the quality of the solution. Firstly we look at the +freely available software package LAPACK \cite{Ande99}, and +then at an example of a commercial software library, the NAG Library +\cite{NAGa05}. The author of this report has to declare an interest in both +of these software products; he is one of the authors of LAPACK and is +currently a software developer employed by NAG Ltd. Naturally, the +examples are chosen because of familiarity with the products and +belief in them as quality products, but I have nevertheless tried not +to introduce bias. + +LAPACK stands for Linear Algebra PACKage and is a numerical software +package for the solution of dense and banded linear algebra problems +aimed at PCs, workstations and high-performance shared memory +machines. One of the aims of LAPACK was to make the software efficient +on modern machines, whilst retaining portability, and to this end it +makes extensive use of the Basic Linear Algebra Subprograms (BLAS), +using block-partitioned algorithms based upon the Level 3 BLAS +wherever possible. The BLAS specify the interface for a set of +subprograms for common scalar and vector (Level 1), matrix-vector +(Level 2) and matrix-matrix operations (Level 3). Their motivation +and specification are given in \cite{Laws79}, \cite{Dong88a} +\cite{Dong90} respectively. Information on +block-partitioned algorithms and performance of LAPACK can be found in +[\cite{Ande99}, Chapter 3]. See also \cite{Golu96}, +particularly Section 1.3, and \cite{Stew98}, Chapter 2, which also +has some nice discussion on computation. + +LAPACK has routines for the solution of systems of linear equations, +linear least squares problems, eigenvalue and singular value problems, +including generalized problems, as well as routines for the underlying +computational components such as matrix factorizations. In addition, a +lot of effort was expended in providing condition and error +estimates. Quoting from the first paragraph of Chapter 4 – Accuracy +and Stability – of the LAPACK Users’ Guide: + +\begin{quote} +In addition to provide faster routines than previously available. LAPACK +provides more comprehensive and better error bounds. Our goal is to provide +error bounds for most quantities computed by LAPACK. +\end{quote} + +In many cases the routines return the bounds directly; in other cases +the Users' Guide gives details of error bounds and provides code +fragments to compute the bounds. + +As an example, routine DGESVX\footnote{In the LAPACK naming scheme the +D stands for double precision, GE for general matrix, SV for solver and +X for expert driver} +solves a system of linear equations +$AX = B$, where $B$ is a matrix of one or more right-hand sides, using +Gaussian elimination with partial pivoting. Part of the interface is +\begin{verbatim} + SUBROUTINE DGESVX(..., RCOND, FERR, BERR, WORK, ..., INFO) +\end{verbatim} +where the displayed arguments return the following information: +\begin{itemize} +\item {\tt RCOND} - Estimate of reciprocal of condition number, +$1/\kappa(A)$ +\item {\tt FERR($j$)} - Estimated forward error for $X_j$ +\item {\tt BERR($j$)} - Componentwise relative backward error for $X_j$ +(smallest relative change in any element of $A$ and $B_j$ that makes $X_j$ +an exact solution) +\item {\tt WORK(l)} - Reciprocal of pivot growth factor, $1/g$ +\item {\tt INFO} - Returns a positive value if the computed triangular +factor $U$ is singular or nearly singular. +\end{itemize} + +Thus DGESVX is returning all the information necessary to judge the +quality of the computed solution. + +The routine returns an estimate of $1/\kappa(A)$, rather than +$\kappa(A)$ to avoid +overflow when $A$ is singular, or very ill-conditioned. The argument +INFO is the LAPACK warning or error flag, and is present in all the +LAPACK user callable routines. It returns zero on successful exit, a +negative value if an input argument is incorrectly supplied, for +example $n < 0$, and a positive value in the case of failure, or near +failure as above. In the above example, INFO returns the value $i$ if +$u_{ii} = 0$, in which case no solution is computed since $U$ is exactly +singular, but returns the value $n + 1$ if $1/\kappa(A) < u$, +in which case $A$ +is non-singular to working precision. In the latter case a solution is +returned, and so INFO = $n + 1$ acts as a warning that the solution may +have no correct digits. The routine also has the option to +equilibrate the matrix $A$. See the documentation of the routine for +further information, either in the Users’ Guide, or in the source code +available from netlib at {\tt http://www.netlib.org/lapack/index.html}. + +As a second example from LAPACK, routine DGEEVX solves the +eigenproblem $Ax = \lambda x$ for the eigenvalues and eigenvectors, +$\lambda_i, x_i, i = 1,2,\ldots,n$ of the $n$ by $n$ matrix $A$. +Optionally, the matrix can be balanced and the left eigenvectors of +$A$ can also be computed. Part of the interface is +\begin{verbatim} + SUBROUTINE DGEEVX(..., ABNRM, RCONDE, RCONDV, ...) +\end{verbatim} +where the displayed arguments return the following information: +\begin{itemize} +\item {\tt ABNRM} - Norm of the balanced matrix +\item {\tt RCONDE($i$)} - Reciprocal of the condition number of the $i$th +eigenvalue, $s_i$ +\item {\tt RCONDV($i$)} - Reciprocal of the condition number for the $i$th +eignevector, sep$_i$ +\end{itemize} + +Following a call to DGEEVX, approximate error bounds for the computed +eigenvalues and eigen- vectors, say EERRBD($i$) and VERRBD($i$), such that +\[|\tilde{\lambda_i}-\lambda_i| \le {\tt EERRBD}(i)\] +\[\theta(\tilde{\nu}_i,\nu_i) \le {\tt VERRBD}(i)\] +where $\theta(\tilde{\nu}_i,\nu_i)$ is the angle between the computed and +true eigenvector, may be returned by the following code fragment, taken +from the Users' Guide: +\begin{verbatim} + EPSMCH = DLAMCH('E') + DO 10 I = 1, N + EERRBD(I) = EPSMCH*ABNRM/RCONDE(I) + VERRBD(I) = EPSMCH*ABNRM/RCONDV(I) + 10 CONTINUE +\end{verbatim} + +These bounds are based upon Table 3, extracted from Table 4.5 of the +LAPACK Users’ Guide, which gives approximate asymptotic error bounds +for the nonsymmetric eigenproblem. These bounds +\begin{center} +\begin{tabular}{|l|c|} +\hline +Simple eigenvalue & $|\tilde{\lambda_i}-\lambda_i|\lesssim\Vert E\Vert_2/s_i$\\ +\hline +Eigenvector & $\theta(\tilde{\nu_i},\nu_i)\lesssim\Vert E\Vert_F/{\rm sep}_i$\\ +\hline +\end{tabular} +\end{center} + +assume that the eigenvalues are simple eigenvalues. In addition if the +problem is ill-conditioned, these bounds may only hold for extremely +small $\Vert E\Vert_2$ and so the Users’ Guide also provides a table of global +error bounds which are not so restrictive on $\Vert E\Vert_2$. +The tables in the +Users’ Guide include bounds for clusters of eigenvalues and for +invariant subspaces, and these bounds can be estimated using DGEESX in +place of DGEEVX. For further details see The LAPACK Users’ Guide +[\cite{Ande99}, Chapter 4] and for further information see +\cite{Golu96}, Chapter 7 and \cite{Stew90}. + +LAPACK is freely available via +netlib\footnote{{\tt http://www.netlib.org/lapack/index.html}}, +is included in the NAG +Fortran 77 Library and is the basis of the dense linear algebra in the +NAG Fortran 90 and C Libraries. Tuned versions of a number of LAPACK +routines are included in the NAG Fortran SMP Library. The matrix +computations of MATLAB have been based upon LAPACK since Version 6 +\cite{Math05}; \cite{High05}. + +We now take an example from the NAG Fortran Library. Routine D01AJF is +a general purpose integrator using an adaptive procedure, based on the +QUADPACK routine QAGS \cite{Pies83}, which performs the +integration +\[I=\int_a^b{f(x)~dx}\] +where $[a,b]$ is a finite interval. Part of the interface to D01AJF is +\begin{verbatim} + SUBROUTINE D01AJF(..., EPSABS, EPSREL, RESULT, ABSERR, ...) +\end{verbatim} +\begin{itemize} +\item {\tt EPSABS} - The absolute accuracy required +\item {\tt EPSREL} - The relative accuracy required +\item {\tt RESULT} - The computed approximation to $I$ +\item {\tt ABSERR} - An estimate of the absolute error +\end{itemize} + +In normal circumstances {\tt ABSERR} satisfies +\[|I-RESULT| \le ABSERR \le {\rm max}(EPSABS,EPSREL\times|I|)\] + +See the NAG Library documentation \cite{NAGb05} and \cite{Pies83} +for further details. QUADPACK is freely available from +netlib,\footnote{{\tt http://www.netlib.org/quadpack}} +and a Fortran 90 version of QAGS is available from the more recent +quadature package, CUBPACK \cite{Cool03}, which is also +available from netlib. Typically the error estimate for a quadrature +routine is obtained at the expense of additional computation with a +finer interval, or mesh, or the use of a higher order quadrature +formula. + +As a second example from the NAG Library we consider the solution of +an ODE. Routine D02PCF integrates +\[y^{\prime}=f(t,y),~{\rm\ given\ }y(t_0)=y_0\] + +where y is the n element solution vector and t is the independent +variable, using a Runge-Kutta method. Following the use of D02PCF, +routine D02PZF may be used to compute global error estimates. Part of +the interface to D02PZF is +\begin{verbatim} + SUBROUTINE D02PZF(RMSERR, ERRMAX, TERRMX, ...) +\end{verbatim} + +where the displayed arguments return the following information: +\begin{itemize} +\item {\tt RMSERR($i$)} - Approximate root mean square error for $y_i$ +\item {\tt ERRMAX} - Maximum approximate true error +\item {\tt TERRMX} - First point at which maximum approximate true error +occurred +\end{itemize} + +The assessment of the error is determined at the expense of computing +a more accurate solution using a higher order method to that used for +the original solution. + +The NAG D02P routines are based upon the RKSUITE software by \cite{Bran92}, +which is also available from +netlib.\footnote{{tt http://www.netlib.org/ode/rksuite/}} See also +\cite{Sham92} and \cite{Bran93}. A Fortran 90 +version of RKSUITE is also +available,\footnote{{\tt http://www.netlib.org/ode/rksuite/} or +{\tt http://www.netlib.org/toms/771}} +see \cite{Bran97}. + +Many routines in the NAG Library attempt to return information about +accuracy. The documentation of the routines includes a section +labelled ``Accuracy'' which, when appropriate, gives further advice or +information. For instance, the optimization routines generally quote +the optimality conditions that need to be met for the routine to be +successful. These routines are cautious, and sometimes return a +warning, or error, when it is likely that an optimum point has been +found, but not all the optimality conditions have been met. NAG and +the authors of the routines feel that this is much the best approach +for reliability – even if users would sometimes prefer that we were +more optimistic! + +\section{Other Approaches} + +What does one do if the software does not provide suitable estimates +for the accuracy of the solution, or the sensitivity of the problem? +One approach is to run the problem with perturbed data and compare +solutions. Of course, the difficulty with this approach is to know how +best to choose perturbations. If a small perturbation does +significantly change the solution, then we can be sure that the +problem is sensitive, but of course we cannot rely on the converse. If +we can have trust that the software implements a stable method, then +any sensitivity in the solution is due to the problem, but otherwise +we cannot be sure whether it is the method or problem that is +sensitive. + +To help estimate such sensitivity there exists software that uses +stochastic methods to give statistical estimates of backward error, +or of sensitivity. One such example, PRECISE, is described in +\cite{Chai96}, Chapter 8 and provides a module +for statistical backward error analysis as well as a module for +sensitivity analysis. Another example is +CADNA\footnote{At the time of writing, a free academic version is +available from {\tt http://www-anp.lip6.fr/cadna/Accueil.php}}; +see for example \cite{Vign93}. + +Another approach to obtaining bounds on the solution is the use of +interval arithmetic, in conjunction with interval analysis \cite{Moor79} +\cite{Krei05}. Some problems can be +successfully solved using interval arithmetic throughout, but for some +problems the bounds obtained would be far too pessimistic; however +interval arithmetic can often be applied as an posteriori tool to +obtain realistic bounds. We note that there is a nice interval +arithmetic toolbox for Matlab, INTLAB, by \cite{Rump99} that is freely +available\footnote{{\tt http://www.ti3.tu-harburg.de/english/index.html}}; +see also \cite{Harg02}. It should be noted that in +general, the aim of interval arithmetic is to return forward error +bounds on the solution. + +\subsection{Example 8.1 (Cancellation and interval arithmetic)} + +{\sl As a very simple example consider the computation of $s$ in Example +3.1 using four figure interval arithmetic. Bearing in mind that +interval arithmetic works with intervals that are guaranteed to +contain the exact solution, we find that} +\[\begin{array}{rcl} +s=[s1~s2]&=&[1.000~1.000]+[1.000\times 10^4~1.000\times 10^4]- +[1.000\times 10^4~1.000\times 10^4]\\ +&=&[1.000\times 10^4~1.001\times 10^4]-[1.000\times 10^4~1.000\times 10^4]\\ +&=&[0~10] +\end{array}\] +{\sl so whilst the result is somewhat pessimistic, it does give due +warning of the cancellation.} + +Finally we comment that one should not be afraid to exert pressure on +software developers to provide features that allow one to estimate +the sensitivity of the problem and the accuracy of the solution. + +\section{Summary} + +We have tried to illustrate the niceties of numerical computation and +the detail that needs to be considered when turning a numerical +algorithm into reliable, robust numerical software. We have also tried +to describe and illustrate the ideas that need to be understood to +judge the quality of a numerical solution, especially condition, +stability and error analysis, including the distinction between +backward and forward errors. + +We emphasise that one should most certainly be concerned about the +quality of computed solutions, and use trustworthy quality +software. We cannot just blithely assume that results returned by +software packages are correct. + +This is not always easy since scientists wish to concentrate on their +science and should not really need to be able to analyse an algorithm +to understand whether or not it is a stable method for solving their +problem. Hence the emphasis in this report on the desirability of +software providing proper measures of the quality of the solution. + +We conclude with a question: +\begin{quote} +``You have been solving these damn problems better than I can pose them''\\ +Sir Edward Bullard, Director NPL, in a remark to Wilkinson in the mid 1950s, +See \cite{Wilk85},p. 11 +\end{quote} + +Software developers should strive to provide solutions that are at least +as good as the data deserves. + \chapter{Chapter Overview} Each routine in the Basic Linear Algebra Subroutine set (BLAS) \cite{blas01} has a prefix where: diff --git a/books/bookvol10.pamphlet b/books/bookvol10.pamphlet index 11b1121..8707a60 100644 --- a/books/bookvol10.pamphlet +++ b/books/bookvol10.pamphlet @@ -19284,6 +19284,64 @@ clean: \end{chunk} \chapter{Algebra Background} +\begin{center} +\includegraphics{ps/sweeney.eps}\\ +"source: Kaisler\cite{Kais09} Complex Adaptive Systems" +\end{center} + +\section{How NAG Libraries were used} + +Based on our experiences with IRENA, we decided to use generic +inter-process communication tools for the link to AXIOM. This has the +added advantage that we can operate across a network. The main +technique we use is the {\sl Remote Procedure Call} (RPC) [Sun +Microsystems Inc., 1988] which allows us to interact with a server on +another machine (or on the local machine). RPC takes care of +differences in data representation (e.g. the byte-order of floating +point numbers) on different architectures. + +AXIOM is a multi-process package. Normally when a user starts up the +system they start up the various components which then interact via +standard socket operations. If they are using the line, they start up +a new process: the NAG Manager (NAGMAN for short). Additionally, there +will be a NAG daemon (NAGD) running on any machine on which the user +may wish to execute NAG routines (which could include the local +host). NAGMAN commnicates with the running AXIOM system via a socket +down which is transmitted the details of and data for the particular +routine to be called. NAGMAN calls a NAGD on another machine via RPC +and eventually returns the results to AXIOM. + +NAGD consists of the server program, and a set of stub codes designed +to call individual NAG routines. It is, in effect, a remotely-callable +version of the NAG library. There is no reason why AXIOM should be the +only system to use it, and indeed there are plans to incorporate the +ability to call NAGD into other systems. + +An ASP is treated just like any other piece of data by the AXIOM-NAG +link. The source code is passed to NAGD and compiled. (There are +various optimisations to prevent the same code being compiled multiple +times, but the details needn't concern us here.) This compiled code is +linked with the NAG Library to make the executable. Thus if a user +calls the same NAG routine with different ASPs the routine will be +relinked each time. + +It would be nice if this were not necessary. The authors of the link +considered two other possibilities: +\begin{itemize} +\item Have AXIOM simulate the ASPs, so that the NAG Library would call +back to AXIOM when it wanted to call an ASP. This was rejected as +being far too slow across a network. +\item Give NAGD the ability to interpret AXIOM or Fortran code. Thus +the NAG routine would call a function which would evaluate a +representation of an ASP to get the required values. This may happen +in the future if data interchange mechanisms between systems are +stanardised, but was rejected for the time being since such a system +would have to be tailored to match each Fortran compiler that NAGD used. +\end{itemize} + +By transmitting source code for ASPs we allow the remote Fortran +compiler to take care of low-level portability problems. + \section{Algebraic Function Fields and Algebraic Geometry} Axiom implements the PAFF package by Hache\cite{Hach95} which deals @@ -20548,8 +20606,92 @@ Given a set of polynomials we'd like to find a 'basis set' (think of the $x-y$ axis in some polynomial space) that is, in some sense, an easier set to use. -\subsection{How To Compute A Basis} +\subsection{How To Compute A Groebner Basis} + +From Verschelde\cite{Vers16} and Norman\cite{Normxx} we have the +algorithm for computing a Groebner Basis. + +Let $I=$. Write $F=\{f_1,f_2,\cdots,f_t\}$. + +{\bf S-polynomials} (Subtraction polynomials) + +A term is a product of a coefficient and a monomial. + +The leading term of a polynomial $p$ (under some monomial ordering, +dicussed below) we will call $LT(p)$. + +The leading monomial of the polynomial we will call $LM(p)$. + +The least common multiple of two monmials $x^a$ and $x^b$ we will +call $LCM(x^a,x^b)$ + +To eliminate the leading term of two nonzero polynomials $p$ and $q$, +we construct the $S$-polynomial +\[S(p,q) = \frac{LCM(LM(p),LM(q))}{LT(p)}\cdot p - +\frac{LCM(LM(p),LM(q))}{LT(q)}\cdot q\] + +If $p$ and $q$ belong to the same ideal $I$ the $S(p,q) \in I$. + +The use of $S$ polynomials to eliminate leading terms of multivariate +polynomials generalizes the row reduction algorithm for systems of +linear equations. If we take a system of homogeneous linear equations +(i.e. the constant coefficient equals zero), then bringing the system +into triangular form yields a Groebner basis. + +{\bf Buchberger's Algorithm:} + +Choose a pair $f_i,f_j$ and compute $\overline{S(f_i,f_j)}^F=h$. +If this is zero, then go to the next pair. If it is not zero, +adjoin $h$ to the set $F$. Then start over with the enlarged $F$ +in place of the original $F$. If the $S$ polynomials are zero +for all pairs, then stop. + +{\sl Example 1} Let $I=$. We use the +lexicographic order with $x > y$. +\begin{enumerate} +\item We compute +\[S(f_1,f_2)=\frac{x^2y}{xy} \cdot (xy-x)- +\frac{x^2y}{x^2} \cdot (x^2-y) = -x^2+y^2\] +\item We compute $\overline{S(f_1,f_2)}^F$. It is $f_3=y^2-y$. +We adjoin this to $F$, so that $F$ is now $F=\{f_1,f_2,f_3\}$. +\item We start over with the new $F$. It is automatic that +$\overline{S(f_i,f_j)}^F=0$. We need to compute +$S(f_1,f_3)$ and $S(f_2,f_3)$. +\item We compute $S(f_1,f_3)=0$ +\item We compute $S(f_2,f_3)=x^3-xy^2$. We compute +$\overline{x^3-xy^2}^F=0$. Note that $F=\{f_1,f_2,f_3\}$. +\end{enumerate} + +We see that $\{f_1,f_2,f_3\}$ is a Groebner basis for the ideal $I$. + +Each step creates a larger set of generators of the ideal $I$ +because we add non-zero elements to our set of generators. + +Eventually we find an element in the ideal whose leading term is +not divisible by any of the leading terms in our set of generators. +We reach a stage where, for every pair of elements $f,g$ we see +that $\overline{S(f,g)}^{F_m}=0$. This means we have found a +Groebner basis. However, this basis is not unique. + +We can compute a unique, minimal Groebner basis by noticing that, +if the leadingTerm(p) is a member of the basis formed by the +leadingTerms of all of the elements of the set when $p$ is +removed, then that smaller set is also a Groebner basis. + +By definition, a Groebner basis $G$ of an ideal $I$ is a minimal +basis provided it satisfies +\begin{enumerate} +\item $leadingCoefficient(p) = 1 \forall p \in G$ +\item $\forall p \in G, the leadingTerm(p) \notin $ +\end{enumerate} +To construct this minimal bases we divide each element in the given +basis by its leading coefficient. Now put the elements in some +arbitrary order. If the first element in $p$ is in $$ +the remove it from $G$. Now go to the second element and perform the +same operation. Once all of the elements are processed the Groebner +basis is minimal (but not unique). Fixing the monomial order will +guarantee a unique reduced Groebner basis. \subsection{Monomial Ordering} Four common monomial orderings are Lexicographic (dictionary), diff --git a/books/bookvol5.pamphlet b/books/bookvol5.pamphlet index 120c87f..da80184 100644 --- a/books/bookvol5.pamphlet +++ b/books/bookvol5.pamphlet @@ -54448,14 +54448,12 @@ We populate the htMacroTable at load time. \chapter{HyperDoc Basic Command support} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Basic Command}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcbasiccommands.eps} -\end{center}} -\vskip 0.5cm +\end{center} + This is the root page of the basic commands dialog. The goal is to present examples of how to construct command lines which demonstrate using Axiom to solve problems. @@ -54464,14 +54462,12 @@ which demonstrate using Axiom to solve problems. %%% HyperDoc -> Basic Commands -> Calculus %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Calculus} -\boxed{\linewidth}{ -\vskip 0.1cm + \begin{center} {\bf {\large{Calculus}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bccalculus.eps} -\end{center}} -\vskip 0.5cm +\end{center} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% HyperDoc -> Basic Commands -> Calculus -> Differentiate @@ -54483,14 +54479,11 @@ which demonstrate using Axiom to solve problems. \calls{bcDifferentiate}{htShowPage} \usesdollar{bcDifferentiate}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Calculus $\rightarrow$ Differentiate}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcdifferentiate.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcDifferentiateGen} due to this line: @@ -54570,14 +54563,11 @@ Pressing the {\bf Continue} calls {\tt bcDifferentiateGen} \calls{bcIndefiniteIntegrate}{htShowPage} \usesdollar{bcIndefiniteIntegrate}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Calculus $\rightarrow$ Indefinite Integral}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcindefiniteintegrate.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcIndefiniteIntegrateGen} due to this line: @@ -54627,14 +54617,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcDefiniteIntegrate}{htShowPage} \usesdollar{bcDefiniteIntegrate}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Calculus $\rightarrow$ Definite Integral}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcdefiniteintegrate.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcDefiniteIntegrateGen} due to this line: @@ -54712,14 +54699,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcLimit}{htShowPage} \usesdollar{bcLimit}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Calculus $\rightarrow$ Limit}}} \vskip 0.25cm \includegraphics[scale=0.75]{ps/v5bclimit.eps} -\end{center}} -\vskip 0.5cm +\end{center} \begin{chunk}{defun bcLimit} (defun |bcLimit| () @@ -54758,14 +54742,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcSum}{htShowPage} \usesdollar{bcSum}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Calculus $\rightarrow$ Summation}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcsum.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcSumGen} due to this line: @@ -54819,27 +54800,23 @@ Pressing the {\bf Continue} button will call the function %%% HyperDoc -> Basic Commands -> Matrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Matrix} -%\boxed{\linewidth}{ -%\vskip 0.1cm + %\begin{center} %{\bf {\large{Calculus}}} %\vskip 0.25cm %\includegraphics[scale=1.0]{ps/v5bccalculus.eps} -%\end{center}} -%\vskip 0.5cm +%\end{center} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% HyperDoc -> Basic Commands -> Matrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \defun{bcMatrix}{Basic Commands - Matrix} -\boxed{\linewidth}{ -\vskip 0.1cm + \begin{center} {\bf {\large{Matrix}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bccalculus.eps} -\end{center}} -\vskip 0.5cm +\end{center} \calls{bcMatrix}{bcReadMatrix} \begin{chunk}{defun bcMatrix} @@ -54851,28 +54828,23 @@ Pressing the {\bf Continue} button will call the function %%% HyperDoc -> Basic Commands -> Draw %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Draw} -%\boxed{\linewidth}{ -%\vskip 0.1cm + %\begin{center} %{\bf {\large{Calculus}}} %\vskip 0.25cm %\includegraphics[scale=1.0]{ps/v5bccalculus.eps} -%\end{center}} -%\vskip 0.5cm +%\end{center} \defun{bcDraw}{Basic Commands - Draw} \calls{bcDraw}{htInitPage} \calls{bcDraw}{bcHt} \calls{bcDraw}{htShowPage} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Draw Basic Command}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcdraw.eps} -\end{center}} -\vskip 0.5cm +\end{center} \begin{chunk}{defun bcDraw} (defun |bcDraw| () @@ -54907,14 +54879,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcDraw2Dfun}{htShowPage} \usesdollar{bcDraw2Dfun}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Draw Basic Command by Function}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcdraw2dfun.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcDraw2DfunGen} due to this line: @@ -54985,14 +54954,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcDraw2Dpar}{htShowPage} \usesdollar{bcDraw2Dpar}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Draw Basic Command by Parameters}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcdraw2dpar.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcDraw2DparGen} due to this line: @@ -55064,14 +55030,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcDraw2DSolve}{htShowPage} \usesdollar{bcDraw2DSolve}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Draw Basic Command by Equation Solution}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcdraw2dsolve.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcDraw2DSolveGen} due to this line: @@ -55146,14 +55109,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcDraw3Dfun}{htShowPage} \usesdollar{bcDraw3Dfun}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Draw Basic Command by 3D function}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcdraw3dfun.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcDraw3DfunGen} due to this line: @@ -55234,14 +55194,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcDraw3Dpar}{htShowPage} \usesdollar{bcDraw3Dpar}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Draw Basic Command by 3D parameterized tube}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcdraw3dpar.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcDraw3DparGen} due to this line: @@ -55318,14 +55275,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcDraw3Dpar1}{htShowPage} \usesdollar{bcDraw3Dpar1}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Draw Basic Command by 3D parameterized function}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcdraw3dpar1.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcDraw3Dpar1Gen} due to this line: @@ -55410,14 +55364,12 @@ Pressing the {\bf Continue} button will call the function %%% HyperDoc -> Basic Commands -> Series %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Series} -%\boxed{\linewidth}{ -%\vskip 0.1cm + %\begin{center} %{\bf {\large{Calculus}}} %\vskip 0.25cm %\includegraphics[scale=1.0]{ps/v5bccalculus.eps} -%\end{center}} -%\vskip 0.5cm +%\end{center} \defun{bcSeries}{Basic Commands - Series} \calls{bcSeries}{htInitPage} @@ -55425,14 +55377,12 @@ Pressing the {\bf Continue} button will call the function \calls{bcSeries}{htShowPage} \usesdollar{bcSeries}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm + \begin{center} {\bf {\large{Matrix Basic Command}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcseries.eps} -\end{center}} -\vskip 0.5cm +\end{center} \begin{chunk}{defun bcSeries} (defun |bcSeries| () @@ -55463,14 +55413,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcSeriesExpansion}{htShowPage} \usesdollar{bcSeriesExpansion}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Series Basic Command expand around a point}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcseriesexpansion.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcSeriesExpansionGen} due to this line: @@ -55531,14 +55478,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcSeriesByFormula}{htMakePage} \calls{bcSeriesByFormula}{htShowPage} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Series Basic Command series by formula}}} \vskip 0.25cm \includegraphics[scale=0.75]{ps/v5bcseriesbyformula.eps} -\end{center}} -\vskip 0.5cm +\end{center} \begin{chunk}{defun bcSeriesByFormula} (defun |bcSeriesByFormula| (a b) @@ -55575,14 +55519,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcTaylorSeries}{htShowPage} \usesdollar{bcTaylorSeries}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Taylor Series Basic Command}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bctaylorseries.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcTaylorSeriesGen} due to this line: @@ -55666,14 +55607,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcLaurentSeries}{htShowPage} \usesdollar{bcLaurentSeries}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Laurent Series Basic Command}}} \vskip 0.25cm \includegraphics[scale=0.8]{ps/v5bclaurentseries.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcLaurentSeriesGen} due to this line: @@ -55741,14 +55679,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcPuiseuxSeries}{htShowPage} \usesdollar{bcPuiseuxSeries}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Puiseux Series Basic Command}}} \vskip 0.25cm \includegraphics[scale=0.75]{ps/v5bcpuiseuxseries.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcPuiseuxSeriesGen} due to this line: @@ -55817,14 +55752,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcSolve}{htShowPage} \usesdollar{bcSolve}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Solve Basic Command}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcsolve.eps} -\end{center}} -\vskip 0.5cm +\end{center} \begin{chunk}{defun bcSolve} (defun |bcSolve| () @@ -55857,14 +55789,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcLinearSolve}{htShowPage} \usesdollar{bcLinearSolve}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Linear Solve Basic Command}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bclinearsolve.eps} -\end{center}} -\vskip 0.5cm +\end{center} \begin{chunk}{defun bcLinearSolve} (defun |bcLinearSolve| (p nn) @@ -55902,14 +55831,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcLinearSolveEqns}{htShowPage} \usesdollar{bcLinearSolveEqns}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Linear Solve Equations Basic Command}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bclinearsolveeqns.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcLinearSolveEqns1} due to this line: @@ -55965,14 +55891,12 @@ after the matrix has been read. \calls{bcReadMatrix}{htMakePage} \calls{bcReadMatrix}{htShowPage} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Matrix Basic Command}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcreadmatrix.eps} -\end{center}} -\vskip 0.5cm +\end{center} + This routine is called from several places to enter a matrix. The argument {\bf bcReadMatrix} is the name of a function to call when the matrix has been entered. This value is set as an {\bf exitFunction} @@ -56025,14 +55949,11 @@ in the page's association table. \usesdollar{bcInputExplicitMatrix}{EmptyMode} \usesdollar{bcInputExplicitMatrix}{bcParseOnly} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Input Explicit Matrix}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcinputexplicitmatrix.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcGenExplicitMatrix} due to this line: @@ -56104,14 +56025,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcInputMatrixByFormula}{htShowPage} \usesdollar{bcInputMatrixByFormula}{bcParseOnly} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Input Matrix By Formula}}} \vskip 0.5cm \includegraphics[scale=1.0]{ps/v5bcinputmatrixbyformula.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcInputMatrixByFormulaGen} due to this line: @@ -56163,14 +56081,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcSystemSolve}{htMakeDoneButton} \calls{bcSystemSolve}{htShowPage} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Solve Directly As Equations}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcsystemsolve.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcSystemSolveEqns1} due to this line: @@ -56409,14 +56324,11 @@ Pressing the {\bf Continue} button will call the function \section{Handling Axiom command execution} -%\boxed{\linewidth}{ -%\vskip 0.1cm %\begin{center} %{\bf {\large{Basic Command}}} %\vskip 0.25cm %\includegraphics[scale=1.0]{ps/v5bcbasiccommands.eps} -%\end{center}} -%\vskip 0.5cm +%\end{center} The {\tt bcGen} function is called with a string which will be passed to the Axiom command line. For example, the path @@ -56650,8 +56562,8 @@ title and a function to call when pressed. \calls{bcProduct}{htShowPage} \usesdollar{bcProduct}{EmptyMode} %TPDHERE see buglist todo 331 -%\begin{minipage}{\linewidth} -% \makebox[\linewidth] +%\begin{minipage}{\textwidth} +% \makebox[\textwidth] % {\includegraphics[scale=0.75]{ps/v5bcproduct.eps}} %\end{minipage} %Pressing the {\bf Continue} button will call the function @@ -56719,14 +56631,11 @@ title and a function to call when pressed. \calls{bcRealLimit}{htShowPage} \usesdollar{bcRealLimit}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Real Limit Basic Command}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcreallimit.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcRealLimitGen} due to this line: @@ -56773,14 +56682,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcRealLimitGen}{htpSetProperty} \calls{bcRealLimitGen}{htShowPage} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Real Limit Basic Command options}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bcreallimitgen.eps} -\end{center}} -\vskip 0.5cm +\end{center} \begin{chunk}{defun bcRealLimitGen} (defun |bcRealLimitGen| (htPage) @@ -56836,14 +56742,11 @@ Pressing the {\bf Continue} button will call the function \calls{bcComplexLimit}{htShowPage} \usesdollar{bcComplexLimit}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Complex Limit Basic Command}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bccomplexlimit.eps} -\end{center}} -\vskip 0.5cm +\end{center} Pressing the {\bf Continue} button will call the function {\bf bcComplexLimitGen} due to this line: @@ -57172,14 +57075,11 @@ If {\bf exitFunction} is set, call it. \calls{bcLinearSolveMatrix1}{htShowPage} \usesdollar{bcLinearSolveMatrix1}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Linear Solve Basic Command options}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bclinearsolvematrix1.eps} -\end{center}} -\vskip 0.5cm +\end{center} \begin{chunk}{defun bcLinearSolveMatrix1} (defun |bcLinearSolveMatrix1| (htPage) @@ -57215,14 +57115,11 @@ If {\bf exitFunction} is set, call it. \calls{bcLinearSolveMatrixInhomo}{htShowPage} \usesdollar{bcLinearSolveMatrixInhomo}{EmptyMode} -\boxed{\linewidth}{ -\vskip 0.1cm \begin{center} {\bf {\large{Linear Solve Basic Command Inhomogeneous}}} \vskip 0.25cm \includegraphics[scale=1.0]{ps/v5bclinearsolvematrixinhomo.eps} -\end{center}} -\vskip 0.5cm +\end{center} \begin{chunk}{defun bcLinearSolveMatrixInhomo} (defun |bcLinearSolveMatrixInhomo| (htPage junk) diff --git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet index 1d6ca21..f1aab33 100644 --- a/books/bookvolbib.pamphlet +++ b/books/bookvolbib.pamphlet @@ -2488,6 +2488,32 @@ when shown in factored form. \section{Numerical Algorithms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\index{Acton, F.S.} +\begin{chunk}{axiom.bib} +@book{Acto70, + author = "Acton, F.S.", + title = "Numerical Methods that (Usually) Work", + year = "1970", + publisher = "Harper and Row", + address = "New York, USA" +} + +\end{chunk} + +\index{Acton, F.S.} +\begin{chunk}{axiom.bib} +@book{Acto96, + author = "Acton, F.S.", + title = "Real Computing Made Real: Preventing Errors in Scientific + and Engineering Calculations", + year = "1996", + publisher = "Princeton University Press", + address = "Princeton, N.J. USA", + isbn = "0-691-03663-2" +} + +\end{chunk} + \index{Ahrens, Peter} \index{Nguyen, Hong Diep} \index{Demmel, James} @@ -2532,6 +2558,20 @@ when shown in factored form. \end{chunk} +\index{Alefeld, G.} +\index{Mayer, G.} +\begin{chunk}{axiom.bib} +@article{Alef00, + author = "Alefeld, G. and Mayer, G.", + title = "Interval analysis: Theory and applications", + journal = "J. Comput. Appl. Math.", + volume = "121", + pages = "421-464", + year = "2000" +} + +\end{chunk} + \index{Anda, A.A.} \index{Park,H.} \begin{chunk}{axiom.bib} @@ -2598,6 +2638,53 @@ when shown in factored form. \end{chunk} +\index{Anderson, E.} +\index{Bai, Z.} +\index{Bischof, C.} +\index{Blackford, S.} +\index{Demmel, J.} +\index{Dongarra, J.} +\index{Du Croz, J.} +\index{Greenbaum, A.} +\index{Hammarling, S.} +\index{McKenney, A.} +\index{Sorensen, D.} +\begin{chunk}{axiom.bib} +@misc{LAPA99, + author = "Anderson E. et al.", + title = "LAPACK User's Guide Third Addition", + year = "1999", + month = "August", + url = "http://www.netlib.org/lapack/lug/" +} + +\end{chunk} + +\index{Anderson, E.} +\index{Bai, Z.} +\index{Bischof, S.} +\index{Blackford, S.} +\index{Demmel, J.} +\index{Dongarra, J. J.} +\index{DuCroz, J.} +\index{Greenbaum, A.} +\index{Hammarling, S.} +\index{McKenney, A.} +\index{Sorensen, D. C.} +\begin{chunk}{axiom.bib} +@book{Ande99, + author = "Anderson, E. and Bai, Z. and Bischof, S. and Blackford, S. and + Demmel, J. and Dongarra, J. J. and DuCroz, J. and Greenbaum, A. + and Hammarling, S. and McKenney, A. Sorensen, D. C.", + title = "LAPACK Users' Guide", + publisher = "SIAM", + year = "1999", + isbn = "0-89871-447-8", + url = "www.netlib.org/lapack/lug/" +} + +\end{chunk} + \index{Baudin, Michael} \index{Smith, Robert L.} \begin{chunk}{axiom.bib} @@ -2621,6 +2708,47 @@ when shown in factored form. \end{chunk} +\index{Bindel, D.} +\index{Demmel, J.} +\index{Kahan, W.} +\index{Marques, O.} +\begin{chunk}{axiom.bib} +@article{Bind02, + author = "Bindel, D. and Demmel, J. and Kahan, W. and Marques, O.", + title = "On computing Givens rotations reliably and efficiently", + journal = "ACM Trans. Math. Software", + volume = "28", + pages = "206-238", + year = "2002" +} + +\end{chunk} + +\index{Blackford, L. S.} +\index{Cleary, A.} +\index{Demmel, J.} +\index{Dhillon, I.} +\index{Dongarra, J. J.} +\index{Hammarling, S.} +\index{Petitet, A.} +\index{Ren, H.} +\index{Stanley, K.} +\index{Whaley, R. C.} +\begin{chunk}{axiom.bib} +@article{Blac97, + author = "Blackford, L. S. and Cleary, A. and Demmel, J. and Dhillon, I. + and Dongarra, J. J. and Hammarling, S. and Petitet, A. and + Ren, H. and Stanley, K. and Whaley, R. C.", + title = "Practical experience in the numerical dangers of heterogeneous + computing", + journal = "ACM Trans. Math. Software", + volume = "23", + pages = "133-147", + year = "1997" +} + +\end{chunk} + \index{Boisvert, Ronald F.} \index{Pozo, Roldan} \index{Remington, Karin A.} @@ -2646,6 +2774,64 @@ when shown in factored form. \end{chunk} +\index{Brankin, R. W.} +\index{Gladwell, I.} +\begin{chunk}{axiom.bib} +@article{Bran97, + author = "Brankin, R. W. and Gladwell, I.", + title = "rksuite\_90: Fortran 90 software for ordinary differential + equation initial-value problems", + journal = "ACM Trans. Math. Software", + volume = "23", + pages = "402-415", + year = "1997" +} + +\end{chunk} + +\index{Brankin, R. W.} +\index{Gladwell, I.} +\index{Shampine, L. F.} +\begin{chunk}{axiom.bib} +@techreport{Bran92, + author = "Brankin, R. W. and Gladwell, I. and Shampine, L. F.", + title = "RKSUITE: A suite of runge-kutta codes for the initial value + problem for ODEs", + year = "1992", + institution = "Southern Methodist University, Dept of Math.", + number = "Softreport 92-S1", + type = "Technical Report" +} + +\end{chunk} + +\index{Brankin, R. W.} +\index{Gladwell, I.} +\index{Shampine, L. F.} +\begin{chunk}{axiom.bib} +@article{Bran93, + author = "Brankin, R. W. and Gladwell, I. and Shampine, L. F.", + title = "RKSUITE: A suite of explicit runge-kutta codes", + journal = "Contributions ot Numerical Mathematics", + pages = "41-53", + publisher = "World Scientific", + year = "1993" +} + +\end{chunk} + +\index{Britton, J. L.} +\begin{chunk}{axiom.bib} +@book{Brit92, + author = "Britton, J. L.", + title = "Collected Works of A. M. Turing: Pure Mathematics", + publisher = "North-Holland", + year = "1992", + isbn = "0-444-88059-3" +} + +\end{chunk} + \index{Bronstein, Manuel} \begin{chunk}{axiom.bib} @misc{Bron99, @@ -2712,6 +2898,141 @@ when shown in factored form. \end{chunk} +\index{Chaitin-Chatelin, F.} +\index{Fraysse, V.} +\begin{chunk}{axiom.bib} +@book{Chai96, + author = "Chaitin-Chatelin, F. and Fraysse, V.", + title = "Lectures on Finite Precision Computations", + publisher = "SIAM", + year = "1996", + isbn = "0-89871-358-7" +} + +\end{chunk} + +\index{Chan, T. F.} +\index{Golub, G. H.} +\index{LeVeque, R. J.} +\begin{chunk}{axiom.bib} +@article{Chan83, + author = "Chan, T. F. and Golub, G. H. and LeVeque, R. J.", + title = "Algorithms for computing the sample variance: Analysis and + recommendations", + journal = "The American Statistician", + volume = "37", + pages = "242-247", + year = "1983" +} + +\end{chunk} + +\index{Cools, R.} +\index{Haegemans, A.} +\begin{chunk}{axiom.bib} +@article{Cool03, + author = "Cools, R. and Haegemans, A.", + title = "Algorithm 824: CUBPACK: A package for automatic cubature; + framework description", + journal = "ACM Trans. Math. Software", + volume = "29", + pages = "287-296", + year = "2003" +} + +\end{chunk} + +\index{Cox, M. G.} +\index{Dainton, M. P.} +\index{Harris, P. M.} +\begin{chunk}{axiom.bib} +@techreport{Coxx00, + author = "Cox, M. G. and Dainton, M. P. and Harris, P. M.", + title = "Testing spreadsheets and other packages used in metrology: + Testing functions for the calculation of standard deviation", + year = "2000", + institution = "National Physical Lab, Teddington, Middlesex UK", + type = "Technical Report", + number = "NPL Report CMSC07/00" +} + +\end{chunk} + +\index{Davis, Timothy A.} +\index{Hu, Yifan} +\begin{chunk}{axiom.bib} +@article{Davi11, + author = "Davis, Timothy A. and Hu, Yifan", + title = "The University of Florida Sparse Matrix Collection", + journal = "ACM Trans. on Math. Software", + volume = "38", + number = "1", + year = "2011", + month = "November", + url = "http://yifanhu.net/PUB/matrices.pdf", + abstract = + "We describe the Univerity of Florida Sparse Matrix Collection, a large + and actively growing set of sparse matrices that arise in real + applications. The Collection is widely used by the numerical linear + algebra community for the development and performance evaluation of + sparse matrix algorithms. It allows for robust and repeatable + experiments: robust because performance results with artificially + generated matrices can be misleading, and repeatable because matrices + are curated and made publicly available in many formats. Its matrices + cover a wide spectrum of domains, including those arising from + problems with underlying 2D or 3D geometry (as structural engineering, + computational fluid dynamics, model reduction, electromagnetics, + semiconductor devices, thermodynamics, materials, acoustics, computer + graphics/vision, robitics/kinematics, and other discretizations) and + those that typically do not have such geometry (optimization, circuit + simulation, economic and financial modeling, theoretical and quantum + chemistry, chemical process simulation, mathematics and statistics, + power networks, and other networks and graphs). We provide software + for accessing and managing the Collection, from MATLAB, Mathematica, + Fortran, and C, as well as an online search capability. Graph + visualization of the matrices is provided, and a new multilevel + coarsening scheme is proposed to facilitate this task.", + paper = "Davi11.pdf" +} + +\end{chunk} + +\index{Davis, Timothy} +\index{Rajamanickam, Sivasankaran} +\index{Sid-Lakhdar, Wissam M.} +\begin{chunk}{axiom.bib} +@techreport{Davi16, + author = "Davis, Timothy and Rajamanickam, Sivasankaran and + Sid-Lakhdar, Wissam M.", + title = "A survey of direct methods for sparse linear systems", + year = "2016", + month = "April", + institution = "Texas A and M", + type = "Technical Report", + url = +"http://faculty.cse.tamu.edu/davis/publications_files/survey_tech_report.pdf", + abstract = + "Wilkinson defined a sparse matrix as one with enough zeros that it + pays to take advantage of them. This informal yet practical definition + captures the essence of the goal of direct methods for solving sparse + matrix problems. They exploit the sparsity of a matrix to solve + problems economically: much faster and using far less memory than if + all the entries of a matrix were stored and took part in explicit + computations. These methods form the backbone of a wide range of + problems in computational science. A glimpse of the breadth of + applications relying on sparse solvers can be seen in the origins of + matrices in published matrix benchmark collections. The goal of this + survey article is to impart a working knowledge of the underlying + theory and practice of sparse direct methods for solving linear + systems and least-squares problems, and to provide an overview of the + algorithms, data structures, and software available to solve these + problems, so that the reader can both understand the methods and know + how best to use them.", + paper = "Davi16.pdf" +} + +\end{chunk} + \index{Demmel, James} \index{Kahan, W.} \begin{chunk}{axiom.bib} @@ -2821,93 +3142,18 @@ when shown in factored form. \end{chunk} -\index{Davis, Timothy A.} -\index{Hu, Yifan} +\index{Demmel, James} +\index{Kahan, W.} \begin{chunk}{axiom.bib} -@article{Davi11, - author = "Davis, Timothy A. and Hu, Yifan", - title = "The University of Florida Sparse Matrix Collection", - journal = "ACM Trans. on Math. Software", - volume = "38", - number = "1", - year = "2011", - month = "November", - url = "http://yifanhu.net/PUB/matrices.pdf", - abstract = - "We describe the Univerity of Florida Sparse Matrix Collection, a large - and actively growing set of sparse matrices that arise in real - applications. The Collection is widely used by the numerical linear - algebra community for the development and performance evaluation of - sparse matrix algorithms. It allows for robust and repeatable - experiments: robust because performance results with artificially - generated matrices can be misleading, and repeatable because matrices - are curated and made publicly available in many formats. Its matrices - cover a wide spectrum of domains, including those arising from - problems with underlying 2D or 3D geometry (as structural engineering, - computational fluid dynamics, model reduction, electromagnetics, - semiconductor devices, thermodynamics, materials, acoustics, computer - graphics/vision, robitics/kinematics, and other discretizations) and - those that typically do not have such geometry (optimization, circuit - simulation, economic and financial modeling, theoretical and quantum - chemistry, chemical process simulation, mathematics and statistics, - power networks, and other networks and graphs). We provide software - for accessing and managing the Collection, from MATLAB, Mathematica, - Fortran, and C, as well as an online search capability. Graph - visualization of the matrices is provided, and a new multilevel - coarsening scheme is proposed to facilitate this task.", - paper = "Davi11.pdf" -} - -\end{chunk} - -\index{Davis, Timothy} -\index{Rajamanickam, Sivasankaran} -\index{Sid-Lakhdar, Wissam M.} -\begin{chunk}{axiom.bib} -@techreport{Davi16, - author = "Davis, Timothy and Rajamanickam, Sivasankaran and - Sid-Lakhdar, Wissam M.", - title = "A survey of direct methods for sparse linear systems", - year = "2016", - month = "April", - institution = "Texas A and M", - type = "Technical Report", - url = -"http://faculty.cse.tamu.edu/davis/publications_files/survey_tech_report.pdf", - abstract = - "Wilkinson defined a sparse matrix as one with enough zeros that it - pays to take advantage of them. This informal yet practical definition - captures the essence of the goal of direct methods for solving sparse - matrix problems. They exploit the sparsity of a matrix to solve - problems economically: much faster and using far less memory than if - all the entries of a matrix were stored and took part in explicit - computations. These methods form the backbone of a wide range of - problems in computational science. A glimpse of the breadth of - applications relying on sparse solvers can be seen in the origins of - matrices in published matrix benchmark collections. The goal of this - survey article is to impart a working knowledge of the underlying - theory and practice of sparse direct methods for solving linear - systems and least-squares problems, and to provide an overview of the - algorithms, data structures, and software available to solve these - problems, so that the reader can both understand the methods and know - how best to use them.", - paper = "Davi16.pdf" -} - -\end{chunk} - -\index{Demmel, James} -\index{Kahan, W.} -\begin{chunk}{axiom.bib} -@article{Demm90, - author = "Demmel, James and Kahan, W.", - title = "Accurate Singular Values of Bidiagonal Matrices", - journal = "SIAM J. Sci. Stat. Comput.", - volume = "11", - number = "5", - pages = "873-912", - year = "1990", - url = "http://www.netlib.org/lapack/lawnspdf/lawn03.pdf", +@article{Demm90, + author = "Demmel, James and Kahan, W.", + title = "Accurate Singular Values of Bidiagonal Matrices", + journal = "SIAM J. Sci. Stat. Comput.", + volume = "11", + number = "5", + pages = "873-912", + year = "1990", + url = "http://www.netlib.org/lapack/lawnspdf/lawn03.pdf", abstract = "Computing the singular values of a bidiagonal matrix is the final phase of the standard algorithm for the singular value decomposition @@ -2978,6 +3224,208 @@ when shown in factored form. \end{chunk} +\index{Dhillon, Inderjit Singh} +\begin{chunk}{axiom.bib} +@phdthesis{Dhil97, + author = "Dhillon, Inderjit Singh", + title = "A New $O(n^2)$ Algorithm for the Symmetric Tridiagonal + Eigenvalue/Eigenvector Problem", + school = "University of California, Berkeley", + year = "1997", + url = "http://www.eecs.berkeley.edu/Pubs/TechRpts/1997/CSD-97-971.pdf", + abstract = + "Computing the eigenvalues and orthogonal eigenvectors of an $n\times n$ + symmetric tridiagonal matrix is an important task that arises while + solving any symmetric eigenproblem. All practical software requires + $O(n^3)$ time to compute all the eigenvectors and ensure their + orthogonality when eigenvalues are close. In the first part of this + thesis we review earlier work and show how some existing + implementations of inverse iteration can fail in surprising ways. + + The main contribution of this thesis is a new $O(n^2)$, easily + parallelizable algorithm for solving the tridiagonal + eigenproblem. Three main advances lead to our new algorithm. A + tridiagonal matrix is traditionally represented by its diagonal and + off-diagonal elements. Our most important advance is in recognizing + that its bidiagonal factors are ``better'' for computational + purposes. The use of bidiagonals enables us to invoke a relative + criterion to judge when eigenvalues are ``close''. The second advance + comes with using multiple bidiagonal factorizations in order to + compute different eigenvectors independently of each other. Thirdly, + we use carefully chosen dqds-like transformations as inner loops to + compute eigenpairs that are highly accurate and ``faithful'' to the + various bidiagonal representations. Orthogonality of the eigenvectors + is a consequence of this accuracy. Only $O(n)$ work per eigenpair is + neede by our new algorithm. + + Conventional wisdom is that there is usually a trade-off between speed + and accuracy in numerical procedures, i.e., higher accuracy can be + achieved only at the expense of greater computing time. An interesting + aspect of our work is that increased accuracy in the eigenvalues and + eigenvectors obviates the need for explicit orthogonalization and + leads to greater speed. + + We present timing and accuracy results comparing a computer + implementation of our new algorithm with four existing EISPACK and + LAPACK software routines. Our test-bed contains a variety of + tridiagonal matrices, some coming from quantum chemistry + applications. The numerical results demonstrate the superiority of + our new algorithm. For example, on a matrix of order 966 that occurs in + the modeling of a biphenyl molecule our method is about 10 times + faster than LAPACK's inverse iteration on a serial IBM RS/6000 + processor and nearly 100 times faster on a 128 processor IBM SP2 + parallel machine.", + paper = "Dhil97.pdf" +} + +\end{chunk} + +\index{Dhillon, Inderjit S.} +\index{Parlett, Beresford N.} +\begin{chunk}{axiom.bib} +@article{Dhil04a, + author = "Dhillon, Inderjit S. and Parlett, Beresford N.", + title = "Multiple representations to compute orthogonal eigenvectors + of symmetric tridiagonal matrices", + journal = "Linear Algebra and its Applications", + volume = "387", + number ="1", + pages = "1-28", + year = "2004", + month = "August", + abstract = + "In this paper we present an $O(nk)$ procedure, Algorithm $MR^3$, for + computing $k$ eigenvectors of an $n\times n$ symmetric tridiagonal + matrix $T$. A salient feature of the algorithm is that a number of + different $LDL^t$ products ($L$ unit lower triangular, $D$ diagonal) + are computed. In exact arithmetic each $LDL^t$ is a factorization of a + translate of $T$. We call the various $LDL^t$ productions + {\sl representations} (of $T$) and, roughly speaking, there is a + representation for each cluster of close eigenvalues. The unfolding of + the algorithm, for each matrix, is well described by a + {\sl representation tree}. We present the tree and use it to show that if + each representation satisfies three prescribed conditions then the + computed eigenvectors are orthogonal to working accuracy and have + small residual norms with respect to the original matrix $T$.", + paper = "Dhil04a.pdf" +} + +\end{chunk} + +\index{Dhillon, Inderjit S.} +\index{Parlett, Beresford N.} +\begin{chunk}{axiom.bib} +@article{Dhil04, + author = "Dhillon, Inderjit S. and Parlett, Beresford N.", + title = "Orthogonal Eigenvectors and Relative Gaps", + journal = "SIAM Journal on Matrix Analysis and Applications", + volume = "25", + year = "2004", + abstract = + "Let $LDL^t$ be the triangular factorization of a real symmetric + $n\times n$ tridiagonal matrix so that $L$ is a unit lower bidiagonal + matrix, $D$ is diagonal. Let $(\lambda,\nu)$ be an eigenpair, + $\lambda \ne 0$, with the property that both $\lambda$ and $\nu$ are + determined to high relative accuracy by the parameters in $L$ and $D$. + Suppose also that the relative gap between $\lambda$ and its nearest + neighbor $\mu$ in the spectrum exceeds $1/n; n|\lambda-\mu| > |\lambda|$. + + This paper presents a new $O(n)$ algorithm and a proof that, in the + presence of round-off errors, the algorithm computes an approximate + eigenvector $\hat{\nu}$ that is accurate to working precision + $|sin \angle(\nu,\hat{\nu})| = O(n\epsilon)$, where $\epsilon$ is the + round-off unit. It follows that $\hat{\nu}$ is numerically orthogonal to + all the other eigenvectors. This result forms part of a program to + compute numerically orthogonal eigenvectors without resorting to the + Gram-Schmidt process. + + The contents of this paper provide a high-level description and + theoretical justification for LAPACK (version 3.0) subroutine DLAR1V.", + paper = "Dhil04.pdf" +} + +\end{chunk} + +\index{Dodson, D. S.} +\begin{chunk}{axiom.bib} +@article{Dods83, + author = "Dodson, D. S.", + title = "Corrigendum: Remark on 'Algorithm 539: Basic Linear Algebra + Subroutines for FORTRAN usage", + journal = "ACM Trans. Math. Software", + volume = "9", + pages = "140", + year = "1983" +} + +\end{chunk} + +\index{Dodson, D. S.} +\index{Grimes, R. G.} +\begin{chunk}{axiom.bib} +@article{Dods82, + author = "Dodson, D. S. and Grimes, R. G.", + title = "Remark on algorithm 539: Basic Linear Algebra Subprograms for + Fortran usage", + journal = "ACM Trans. Math. Software", + volume = "8", + pages = "403-404", + year = "1982" +} + +\end{chunk} + +\index{Dongarra, J. J.} +\index{DuCroz, J.} +\index{Hammarling, S.} +\index{Hanson, R. J.} +\begin{chunk}{axiom.bib} +@article{Dong88, + author = "Dongarra, J. J. and DuCroz, J. and Hammarling, S. and + Hanson, R. J.", + title = "An extended set of FORTRAN Basic Linear Algebra Subprograms", + journal = "ACM Trans. Math. Software", + volume = "14", + pages = "1-32", + year = "1988" +} + +\end{chunk} + +\index{Dongarra, J. J.} +\index{DuCroz, J.} +\index{Hammarling, S.} +\index{Hanson, R. J.} +\begin{chunk}{axiom.bib} +@article{Dong88a, + author = "Dongarra, J. J. and DuCroz, J. and Hammarling, S. and + Hanson, R. J.", + title = "Corrigenda: 'An extended set of FORTRAN Basic Linear Algebra + Subprograms", + journal = "ACM Trans. Math. Software", + volume = "14", + pages = "399", + year = "1988" +} + +\end{chunk} + +\index{Dongarra, J.} +\index{DuCroz, J.} +\index{Duff, I. S.} +\index{Hammarling, S.} +\begin{chunk}{axiom.bib} +@article{Dong90, + author = "Dongarra, J. and DuCroz, J. and Duff, I. S. and Hammarling, S.", + title = "A set of Level 3 Basic Linear Algebra Subprograms", + journal = "ACM Trans. Math. Software", + volume = "16", + pages = "1-28", + year = "1990" +} + +\end{chunk} + \index{Drmac, Zlatko} \begin{chunk}{axiom.bib} @article{Drma97, @@ -3093,138 +3541,44 @@ when shown in factored form. pages = "1-28", comment = "LAPACK Working note 176", url = "http://www.netlib.org/lapack/lawnspdf/lawn176.pdf", - abstract = - "This note reports an unexpected and rather erratic behavior of the - LAPACK software implementation of the QR factorization with - Businger-Golub column pivoting. It is shown that, due to finite - precision arithmetic, software implementation of the factorization can - catastrophically fail to produce triangular factor with the structure - characteristic to the Businger-Golub pivot strategy. The failure of - current {\sl state of the art} software, and a proposed alternative - implementations are analyzed in detail.", - paper = "Drma08c.pdf" -} - -\end{chunk} - -\index{Dhillon, Inderjit Singh} -\begin{chunk}{axiom.bib} -@phdthesis{Dhil97, - author = "Dhillon, Inderjit Singh", - title = "A New $O(n^2)$ Algorithm for the Symmetric Tridiagonal - Eigenvalue/Eigenvector Problem", - school = "University of California, Berkeley", - year = "1997", - url = "http://www.eecs.berkeley.edu/Pubs/TechRpts/1997/CSD-97-971.pdf", - abstract = - "Computing the eigenvalues and orthogonal eigenvectors of an $n\times n$ - symmetric tridiagonal matrix is an important task that arises while - solving any symmetric eigenproblem. All practical software requires - $O(n^3)$ time to compute all the eigenvectors and ensure their - orthogonality when eigenvalues are close. In the first part of this - thesis we review earlier work and show how some existing - implementations of inverse iteration can fail in surprising ways. - - The main contribution of this thesis is a new $O(n^2)$, easily - parallelizable algorithm for solving the tridiagonal - eigenproblem. Three main advances lead to our new algorithm. A - tridiagonal matrix is traditionally represented by its diagonal and - off-diagonal elements. Our most important advance is in recognizing - that its bidiagonal factors are ``better'' for computational - purposes. The use of bidiagonals enables us to invoke a relative - criterion to judge when eigenvalues are ``close''. The second advance - comes with using multiple bidiagonal factorizations in order to - compute different eigenvectors independently of each other. Thirdly, - we use carefully chosen dqds-like transformations as inner loops to - compute eigenpairs that are highly accurate and ``faithful'' to the - various bidiagonal representations. Orthogonality of the eigenvectors - is a consequence of this accuracy. Only $O(n)$ work per eigenpair is - neede by our new algorithm. - - Conventional wisdom is that there is usually a trade-off between speed - and accuracy in numerical procedures, i.e., higher accuracy can be - achieved only at the expense of greater computing time. An interesting - aspect of our work is that increased accuracy in the eigenvalues and - eigenvectors obviates the need for explicit orthogonalization and - leads to greater speed. - - We present timing and accuracy results comparing a computer - implementation of our new algorithm with four existing EISPACK and - LAPACK software routines. Our test-bed contains a variety of - tridiagonal matrices, some coming from quantum chemistry - applications. The numerical results demonstrate the superiority of - our new algorithm. For example, on a matrix of order 966 that occurs in - the modeling of a biphenyl molecule our method is about 10 times - faster than LAPACK's inverse iteration on a serial IBM RS/6000 - processor and nearly 100 times faster on a 128 processor IBM SP2 - parallel machine.", - paper = "Dhil97.pdf" -} - -\end{chunk} - -\index{Dhillon, Inderjit S.} -\index{Parlett, Beresford N.} -\begin{chunk}{axiom.bib} -@article{Dhil04a, - author = "Dhillon, Inderjit S. and Parlett, Beresford N.", - title = "Multiple representations to compute orthogonal eigenvectors - of symmetric tridiagonal matrices", - journal = "Linear Algebra and its Applications", - volume = "387", - number ="1", - pages = "1-28", - year = "2004", - month = "August", - abstract = - "In this paper we present an $O(nk)$ procedure, Algorithm $MR^3$, for - computing $k$ eigenvectors of an $n\times n$ symmetric tridiagonal - matrix $T$. A salient feature of the algorithm is that a number of - different $LDL^t$ products ($L$ unit lower triangular, $D$ diagonal) - are computed. In exact arithmetic each $LDL^t$ is a factorization of a - translate of $T$. We call the various $LDL^t$ productions - {\sl representations} (of $T$) and, roughly speaking, there is a - representation for each cluster of close eigenvalues. The unfolding of - the algorithm, for each matrix, is well described by a - {\sl representation tree}. We present the tree and use it to show that if - each representation satisfies three prescribed conditions then the - computed eigenvectors are orthogonal to working accuracy and have - small residual norms with respect to the original matrix $T$.", - paper = "Dhil04a.pdf" + abstract = + "This note reports an unexpected and rather erratic behavior of the + LAPACK software implementation of the QR factorization with + Businger-Golub column pivoting. It is shown that, due to finite + precision arithmetic, software implementation of the factorization can + catastrophically fail to produce triangular factor with the structure + characteristic to the Businger-Golub pivot strategy. The failure of + current {\sl state of the art} software, and a proposed alternative + implementations are analyzed in detail.", + paper = "Drma08c.pdf" } \end{chunk} -\index{Dhillon, Inderjit S.} -\index{Parlett, Beresford N.} +\index{Dubrulle, A. A.} \begin{chunk}{axiom.bib} -@article{Dhil04, - author = "Dhillon, Inderjit S. and Parlett, Beresford N.", - title = "Orthogonal Eigenvectors and Relative Gaps", - journal = "SIAM Journal on Matrix Analysis and Applications", - volume = "25", - year = "2004", - abstract = - "Let $LDL^t$ be the triangular factorization of a real symmetric - $n\times n$ tridiagonal matrix so that $L$ is a unit lower bidiagonal - matrix, $D$ is diagonal. Let $(\lambda,\nu)$ be an eigenpair, - $\lambda \ne 0$, with the property that both $\lambda$ and $\nu$ are - determined to high relative accuracy by the parameters in $L$ and $D$. - Suppose also that the relative gap between $\lambda$ and its nearest - neighbor $\mu$ in the spectrum exceeds $1/n; n|\lambda-\mu| > |\lambda|$. +@article{Dubr83, + author = "Dubrulle, A. A.", + title = "A class of numerical methods for the computation of Pythagorean + sums", + journal = "IBM J. Res. Develop.", + volume = "27", + number = "6", + pages = "582-589", + year = "1983" +} - This paper presents a new $O(n)$ algorithm and a proof that, in the - presence of round-off errors, the algorithm computes an approximate - eigenvector $\hat{\nu}$ that is accurate to working precision - $|sin \angle(\nu,\hat{\nu})| = O(n\epsilon)$, where $\epsilon$ is the - round-off unit. It follows that $\hat{\nu}$ is numerically orthogonal to - all the other eigenvectors. This result forms part of a program to - compute numerically orthogonal eigenvectors without resorting to the - Gram-Schmidt process. - - The contents of this paper provide a high-level description and - theoretical justification for LAPACK (version 3.0) subroutine DLAR1V.", - paper = "Dhil04.pdf" +\end{chunk} + +\index{Einarsson, B.} +\begin{chunk}{axiom.bib} +@book{Eina05, + author = "Einarsson, B.", + title = "Accuracy and Reliability in Scientific Computing", + publisher = "SIAM", + year = "2005", + isbn = "0-89871-584-9", + url = "http://www.nsc.liu.se/wg25/book/" } \end{chunk} @@ -3339,6 +3693,46 @@ when shown in factored form. \end{chunk} +\index{Forsythe, G. E.} +\begin{chunk}{axiom.bib} +@article{Fors70, + author = "Forsythe, G. E.", + title = "Pitfalls in computations, or why a math book isn't enough", + journal = "Amer. Math. Monthly", + volume = "9", + pages = "931-995", + year = "1970" +} + +\end{chunk} + +\index{Forsythe, G. E.} +\begin{chunk}{axiom.bib} +@incollection{Fors69, + author = "Forsythe, G. E.", + title = "What is a satisfactory quadratic equation solver", + booktitle = "Constructive Aspects of the Fundamental Theorem of Algebra", + pages = "53-61", + publisher = "Wiley", + year = "1969" +} + +\end{chunk} + +\index{Fox, L.} +\begin{chunk}{axiom.bib} +@article{Foxx71, + author = "Fox, L.", + title = "How to get meaningless answers in scientific computations (and + what to do about it)", + journal = "IMA Bulletin", + volume = "7", + pages = "296-302", + year = "1971" +} + +\end{chunk} + \index{Gentlman, W. Morven} \index{Marovich, Scott B.} \begin{chunk}{axiom.bib} @@ -3366,6 +3760,150 @@ when shown in factored form. \end{chunk} +\index{Givens, W.} +\begin{chunk}{axiom.bib} +@techreport{Give54, + author = "Givens, W.", + title = "Numerical computation of the characteristic values of a real + symmetric matrix", + year = "1954", + institution = "Oak Ridge National Laboratory", + type = "Technical Report", + number = "ORNL-1574" +} + +\end{chunk} + +\index{Golub, G.H.} +\begin{chunk}{axiom.bib} +@article{Golu65, + author = "Golub, G.H.", + title = "Numerical methods for solving linear least squares problems", + journal = "Numer. Math.", + volume = "7", + pages = "206-216", + year = "1965" +} + +\end{chunk} + +\index{Golub, Gene H.} +\index{Van Loan, Charles F.} +\begin{chunk}{axiom.bib} +@book{Golu89, + author = "Golub, Gene H. and Van Loan, Charles F.", + title = "Matrix Computations", + publisher = "Johns Hopkins University Press", + year = "1989", + isbn = "0-8018-3772-3" +} + +\end{chunk} + +\index{Golub, Gene H.} +\index{Van Loan, Charles F.} +\begin{chunk}{axiom.bib} +@book{Golu96, + author = "Golub, Gene H. and Van Loan, Charles F.", + title = "Matrix Computations", + publisher = "Johns Hopkins University Press", + isbn = "978-0-8018-5414-9", + year = "1996" +} + +\end{chunk} + +\index{Hammarling S.} +\begin{chunk}{axiom.bib} +@article{Hamm85, + author = "Hammarling S.", + title = " The Singular Value Decomposition in Multivariate Statistics", + journal = "ACM Signum Newsletter", + volume = "20", + number = "3", + pages = "2--25", + year = "1985" +} + +\end{chunk} + +\index{Hammarling, Sven} +\begin{chunk}{axiom.bib} +@book{Hamm05, + author = "Hammarling, Sven", + title = "An Introduction to the Quality of Computed Solutions", + booktitle = "Accuracy and Reliability in Scientific Computing", + year = "2005", + publisher = "SIAM", + pages = "43-76", + url = "http://eprints.ma.man.ac.uk/101/", + paper = "Hamm05.pdf" +} + +\end{chunk} + +\index{Hargreaves, G.} +\begin{chunk}{axiom.bib} +@mastersthesis{Harg02, + author = "Hargreaves, G.", + title = "Interval analysis in MATLAB", + school = "University of Manchester, Dept. of Mathematics", + year = "2002" +} + +\end{chunk} + +\index{Higham, D. J.} +\index{Higham, N. J.} +\begin{chunk}{axiom.bib} +@book{High05, + author = "Higham, D. J. and Higham, N. J.", + title = "MATLAB Guide", + publisher = "SIAM", + year = "2002", + isbn = "0-89871-521-0" +} + +\end{chunk} + +\index{Higham, Nicholas J.} +\begin{chunk}{axiom.bib} +@article{High88, + author = "Higham, Nicholas J.", + title = "FORTRAN codes for estimating the one-norm of a real or complex + matrix, with applications to condition estimation", + journal = "ACM Trans. Math. Soft", + volume = "14", + number = "4", + pages = "381-396", + year = "1988" +} + +\end{chunk} + +\index{Higham, Nicholas J.} +\begin{chunk}{axiom.bib} +@misc{High98, + author = "Higham, Nicholas J.", + title = "Can you 'count' on your computer?", + url = "http://www.maths.man.ac.uk/~higham/talks/", + year = "1998" +} + +\end{chunk} + +\index{Higham, Nicholas J.} +\begin{chunk}{axiom.bib} +@book{High02, + author = "Higham, Nicholas J.", + title = "Accuracy and stability of numerical algorithms", + publisher = "SIAM", + isbn = "0-89871-521-0", + year = "2002" +} + +\end{chunk} + \index{Higham, Nicholas J.} \begin{chunk}{axiom.bib} @article{High86, @@ -3397,6 +3935,41 @@ when shown in factored form. \end{chunk} +\begin{chunk}{axiom.bib} +@misc{IEEE85, + author = "IEEE", + title = "ANSI/IEEE Standard for Binary Floating Point Arithmetic: + Std 754-1985", + publisher = "IEEE Press", + year = "1985" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@misc{IEEE87, + author = "IEEE", + title = "ANSI/IEEE Standard for Radix Independent Floating Point Arithmetic: + Std 854-1987", + publisher = "IEEE Press", + year = "1987" +} + +\end{chunk} + +\index{Isaacson, E.} +\index{Keller, H. B.} +\begin{chunk}{axiom.bib} +@book{Isaa94, + author = "Isaacson, E. and Keller, H. B.", + title = "Analysis of Numerical Methods", + publisher = "Dover", + year = "1994", + isbn = "0-486-68029-0" +} + +\end{chunk} + \index{Kagstrom, Bo} \index{Westin, L.} \begin{chunk}{axiom.bib} @@ -3612,6 +4185,20 @@ when shown in factored form. \end{chunk} +\index{Kn\"usel, L.} +\begin{chunk}{axiom.bib} +@article{Knus98, + author = {Kn\"usel, L.}, + title = "On the accuracy of statistical distributions in Microsoft + Excel 97", + journal = "Comput. Statist. Data Anal.", + volume = "26", + pages = "375-377", + year = "1998" +} + +\end{chunk} + \index{K\"ohler, Martin} \index{Saak, Jens} \begin{chunk}{axiom.bib} @@ -3641,6 +4228,17 @@ when shown in factored form. \end{chunk} +\index{Kreinovich, V.} +\begin{chunk}{axiom.bib} +@misc{Krei05, + author = "Kreinovich, V.", + title = "Interval cmoputations", + year = "2005", + url = "http://www.cs.utep.edu/interval-comp/" +} + +\end{chunk} + \index{Kuki, Hirondo} \begin{chunk}{axiom.bib} @article{Kuki72a, @@ -3692,24 +4290,27 @@ when shown in factored form. \end{chunk} -\index{Anderson, E.} -\index{Bai, Z.} -\index{Bischof, C.} -\index{Blackford, S.} -\index{Demmel, J.} -\index{Dongarra, J.} -\index{Du Croz, J.} -\index{Greenbaum, A.} -\index{Hammarling, S.} -\index{McKenney, A.} -\index{Sorensen, D.} +\index{Lawson, C. L.} +\index{Hanson, R. J.} \begin{chunk}{axiom.bib} -@misc{LAPA99, - author = "Anderson E. et al.", - title = "LAPACK User's Guide Third Addition", - year = "1999", - month = "August", - url = "http://www.netlib.org/lapack/lug/" +@book{Laws74, + author = "Lawson, C. L. and Hanson, R. J.", + title = "Solving Least Squares Problems", + publisher = "Prentice-Hall", + year = "1974" +} + +\end{chunk} + +\index{Lawson, C. L.} +\index{Hanson, R. J.} +\begin{chunk}{axiom.bib} +@book{Laws95, + author = "Lawson, C. L. and Hanson, R. J.", + title = "Solving Least Squares Problems", + publisher = "SIAM", + isbn = "0-89871-356-0", + year = "1995" } \end{chunk} @@ -3820,6 +4421,100 @@ when shown in factored form. \end{chunk} +\index{Martin, R. S.} +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@article{Mart68, + author = "Martin, R. S. and Wilkinson, J. H.", + title = "Similarity reduction ofa general matrix to Hessenberg form", + journal = "Numer. Math.", + volume = "12", + pages = "349-368", + year = "1968" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@misc{Math05, + author = "MathWorks", + title = "MATLAB", + publisher = "The Mathworks, Inc.", + url = "http://www.mathworks.com" +} + +\end{chunk} + +\index{McCullough, B. D.} +\index{Wilson, B.} +\begin{chunk}{axiom.bib} +@article{Mccu02, + author = "McCullough, B. D. and Wilson, B.", + title = "On the accuracy of statistical procedures in Microsoft Excel + 2000 and Excel XP", + journal = "Comput. Statist. Data Anal.", + volume = "40", + pages = "713-721", + year = "2002" +} + +\end{chunk} + +\index{McCullough, B. D.} +\index{Wilson, B.} +\begin{chunk}{axiom.bib} +@article{Mccu99, + author = "McCullough, B. D. and Wilson, B.", + title = "On the accuracy of statistical procedures in Microsoft Excel 97", + journal = "Comput. Statist. Data Anal.", + volume = "31", + pages = "27-37", + year = "1999" +} + +\end{chunk} + +\index{Metcalf, M.} +\index{Reid, J. K.} +\begin{chunk}{axiom.bib} +@book{Metc96, + author = "Metcalf, M. and Reid, J. K.", + title = "Fortran 90/95 Explained", + publisher = "Oxford University Press", + year = "1996" +} + +\end{chunk} + +\index{Metcalf, M.} +\index{Reid, J. K.} +\index{Cohen, M.} +\begin{chunk}{axiom.bib} +@book{Metc04, + author = "Metcalf, M. and Reid, J. K. and Cohen, M.", + title = "Fortran 95/2003 Explained", + publisher = "Oxford University Press", + year = "2004", + isbn = "0-19-852693-8" +} + +\end{chunk} + +\index{Moler, C.} +\index{Morrison, D.} +\begin{chunk}{axiom.bib} +@article{Mole83, + author = "Moler, C. and Morrison, D.", + title = "Replacing square roots by Pythagorena sums", + journal = "IBM J. Res. Develop.", + volume = "27", + number = "6", + pages = "577-581", + year = "1983" +} + +\end{chunk} + \index{Moler, C.B.} \index{Stewart, G.W.} \begin{chunk}{axiom.bib} @@ -3856,6 +4551,168 @@ when shown in factored form. \end{chunk} +\index{Moore, R. E.} +\begin{chunk}{axiom.bib} +@book{Moor79, + author = "Moore, R. E.", + title = "methods and Applications of Interval Analysis", + publisher = "SIAM", + year = "1979" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@misc{NAGa05, + author = "Numerical Algorithms Group", + title = "The NAG Library", + url = "http://www.nag.co.uk/numeric", + year = "2005" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@misc{NAGb05, + author = "Numerical Algorithms Group", + title = "The NAG Fortran Library Manual", + url = "http://www.nag.co.uk/numeric/fl/manual/html/FLlibrarymanual.asp", + year = "2005" +} + +\end{chunk} + +\index{Overton, M. L.} +\begin{chunk}{axiom.bib} +@book{Over01, + author = "Overton, M. L.", + title = "Numerical Computing with IEEE Floating Point Arithmetic", + publisher = "SIAM", + year = "2001", + isbn = "0-89871-482-6" +} + +\end{chunk} + +\index{Piessens, R.} +\index{de Doncker-Kapenga, E.}, +\index{\"Uberhuber, C. W.} +\index{Kahaner, D. K.} +\begin{chunk}{axiom.bib} +@book{Pies83, + author = {Piessens, R. and de Doncker-Kapenga, E. and \"Uberhuber, C. W. + and Kahaner, D. K.}, + title = "QUADPACK - A Subroutine Package for Automatic Integration", + publisher = "Springer-Verlag", + year = "1983" +} + +\end{chunk} + +\index{Petersen, Kaare Brandt} +\index{Pedersen, Michael Syskind} +\begin{chunk}{axiom.bib} +@misc{Pete12, + author = "Petersen, Kaare Brandt and Pedersen, Michael Syskind", + title = "The Matrix Cookbook", + url = + "http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf", + year = "2012", + month = "November" +} + +\end{chunk} + +\index{Priest, D. M.} +\begin{chunk}{axiom.bib} +@article{Prie04, + author = "Priest, D. M.", + title = "Efficient scaling for complex division", + journal = "ACM Trans. Math. Software", + volume = "30", + pages = "389-401", + year = "2004" +} + +\end{chunk} + +\index{Rump, S. M.} +\begin{chunk}{axiom.bib} +@InProceedings{Rump99, + author = "Rump, S. M.", + title = "INTLAB - INTerval LABoratory", + booktitle = "Developments in Reliable Computing", + pages = "77-104", + publisher = "Kluwer Academic", + year = "1999" +} + +\end{chunk} + +\index{Shampine, L. F.} +\index{Gladwell, I.} +\begin{chunk}{axiom.bib} +@InProceedings{Sham92, + author = "Shampine, L. F. and Gladwell, I.", + title = "The next generation of runge-kutta codes", + booktitle = "Computational Ordinary Differential Equations", + pages = "145-164", + publisher = "Oxford University Press", + year = "1992" +} + +\end{chunk} + +\index{Smith, R. L.} +\begin{chunk}{axiom.bib} +@article{Smit62, + author = "Smith, R. L.", + title = "Algorithm 116: Complex division", + journal = "Communs. Ass. comput. Mach.", + volume = "5", + pages = "435", + year = "1962" +} + +\end{chunk} + +\index{Stewart, G. W.} +\begin{chunk}{axiom.bib} +@book{Stew98, + author = "Stewart, G. W.", + title = "Matrix Algorithms: Basic Decompositions, volume I", + publisher = "SIAM", + year = "1998", + isbn = "0-89871-414-1" +} + +\end{chunk} + +\index{Stewart, G. W.} +\begin{chunk}{axiom.bib} +@article{Stew85, + author = "Stewart, G. W.", + title = "A note on complex division", + journal = "ACM Trans. Math. Software", + volume = "11", + pages = "238-241", + year = "1985" +} + +\end{chunk} + +\index{Stewart, G. W.} +\index{Sun, J.} +\begin{chunk}{axiom.bib} +@book{Stew90, + author = "Stewart, G. W. and Sun, J.", + title = "Matrix Perturbation Theory", + publisher = "Academic Press", + year = "1990" +} + +\end{chunk} + \index{Stoutemyer, David R.} \begin{chunk}{axiom.bib} @article{Stou07, @@ -3881,20 +4738,6 @@ when shown in factored form. \end{chunk} -\index{Petersen, Kaare Brandt} -\index{Pedersen, Michael Syskind} -\begin{chunk}{axiom.bib} -@misc{Pete12, - author = "Petersen, Kaare Brandt and Pedersen, Michael Syskind", - title = "The Matrix Cookbook", - url = - "http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf", - year = "2012", - month = "November" -} - -\end{chunk} - \index{Sutton, Brian D.} \begin{chunk}{axiom.bib} @article{Sutt13, @@ -3926,16 +4769,28 @@ when shown in factored form. \end{chunk} -\index{Yang, Xiang} -\index{Mittal, Rajat} -\begin{chunk}{ignore} -{Yang14, - author ="Yang, Xiang and Mittal, Rajat", - title = "Acceleration of the Jacobi iterative method by factors exceeding 100 - using scheduled relation", - url = -"http://engineering.jhu.edu/fsag/wp-content/uploads/sites/23/2013/10/JCP_revised_WebPost.pdf", - paper = "Yang14.pdf" +\index{Turing, A. M.} +\begin{chunk}{axiom.bib} +@article{Turi48, + author = "Turing, A. M.", + title = "Rounding-off errors in matrix processes", + journal = "Q. J. Mech. Appl. Math.", + volume = "1", + pages = "287-308", + year = "1948" +} + +\end{chunk} + +\index{Vignes, J.} +\begin{chunk}{axiom.bib} +@article{Vign93, + author = "Vignes, J.", + title = "A stochastic arithmetic for reliable scientific computation", + journal = "Math. and Comp. in Sim.", + volume = "25", + pages = "233-261", + year = "1993" } \end{chunk} @@ -3962,6 +4817,121 @@ when shown in factored form. \end{chunk} +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@book{Wilk63, + author = "Wilkinson, J. H.", + title = "Rounding Erroors in Algebraic Processes", + publisher = "HMSO", + series = "Notes on Applied Science, No. 32", + year = "1963" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@book{Wilk65, + author = "Wilkinson, J. H.", + title = "The Algebraic Eigenvalue Problem", + publisher = "Oxford University Press", + year = "1965" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@InProceedings{Wilk84, + author = "Wilkinson, J. H.", + title = "The perfidious polynomial", + booktitle = "Studies in Numerical Analysis", + volume = "24", + chapter = "1", + pages = "1-28", + year = "1984" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@article{Wilk86, + author = "Wilkinson, J. H.", + title = "Error analysis revisited", + journal = "IMA Bulletin", + volume = "22", + pages = "192-200", + year = "1986" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@article{Wilk61, + author = "Wilkinson, J. H.", + title = "Error analysis of diret methods of matrix inversion", + journal = "J. ACM", + volume = "8", + pages = "281-330", + year = "1961" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@article{Wilk85, + author = "Wilkinson, J. H.", + title = "The state of the art in error analysis", + journal = "NAG Newsletter", + volume = "2/85", + pages = "5-28", + year = "1985" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@article{Wilk60, + author = "Wilkinson, J. H.", + title = "Error analysis of floating-point computation", + journal = "Numer. Math.", + volume = "2", + pages = "319-340", + year = "1960" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\index{Reinsch, C.} +\begin{chunk}{axiom.bib} +@book{Wilk71, + author = "Wilkinson, J. H.", + title = "Handbook for Automatic Computation, V2, Linear Algebra", + publisher = "Springer-Verlag", + year = "1971" +} + +\end{chunk} + +\index{Yang, Xiang} +\index{Mittal, Rajat} +\begin{chunk}{ignore} +{Yang14, + author ="Yang, Xiang and Mittal, Rajat", + title = "Acceleration of the Jacobi iterative method by factors exceeding 100 + using scheduled relation", + url = +"http://engineering.jhu.edu/fsag/wp-content/uploads/sites/23/2013/10/JCP_revised_WebPost.pdf", + paper = "Yang14.pdf" +} + +\end{chunk} + \section{Special Functions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \index{Corless, Robert M.} @@ -33123,24 +34093,6 @@ J. of Pure and Applied Algebra, 45, 225-240 (1987) \end{chunk} -\index{Golub, Gene H.} -\index{Van Loan, Charles F.} -\begin{chunk}{ignore} -\bibitem[Golub 89]{GL89} Golub, Gene H.; Van Loan, Charles F. - title = "Matrix Computations", -Johns Hopkins University Press ISBN 0-8018-3772-3 (1989) - -\end{chunk} - -\index{Golub, Gene H.} -\index{Van Loan, Charles F.} -\begin{chunk}{ignore} -\bibitem[Golub 96]{GL96} Golub, Gene H.; Van Loan, Charles F. - title = "Matrix Computations", -Johns Hopkins University Press ISBN 978-0-8018-5414-9 (1996) - -\end{chunk} - \index{Grabmeier, Johannes} \begin{chunk}{axiom.bib} @misc{Grab91a, @@ -33301,29 +34253,6 @@ Reference manual Edition 2.1.1 May 2004 \end{chunk} -\index{Hammarling S.} -\begin{chunk}{ignore} -\bibitem[Hammarling 85]{Ham85} Hammarling S. - title = " The Singular Value Decomposition in Multivariate Statistics", -ACM Signum Newsletter. 20, 3 2--25. (1985) - -\end{chunk} - -\index{Hammarling, Sven} -\begin{chunk}{axiom.bib} -@book{Hamm05, - author = "Hammarling, Sven", - title = "An Introduction to the Quality of Computed Solutions", - booktitle = "Accuracy and Reliability in Scientific Computing", - year = "2005", - publisher = "SIAM", - pages = "43-76", - url = "http://eprints.ma.man.ac.uk/101/", - paper = "Hamm05.pdf" -} - -\end{chunk} - \index{Hammersley, J. M.} \index{Handscomb, D. C.} \begin{chunk}{ignore} @@ -33447,22 +34376,6 @@ January 1956, 10-15 \end{chunk} -\index{Higham, Nicholas J.} -\begin{chunk}{ignore} -\bibitem[Higham 88]{Hig88} Higham, N.J. - title = "FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation", -ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988. - -\end{chunk} - -\index{Higham, Nicholas J.} -\begin{chunk}{ignore} -\bibitem[Higham 02]{Hig02} Higham, Nicholas J. - title = "Accuracy and stability of numerical algorithms", -SIAM Philadelphia, PA ISBN 0-89871-521-0 (2002) - -\end{chunk} - \index{Hock, W.} \index{Schittkowski, K.} \begin{chunk}{ignore} diff --git a/books/bookvolbug.pamphlet b/books/bookvolbug.pamphlet index 6745ff4..2b0a31d 100644 --- a/books/bookvolbug.pamphlet +++ b/books/bookvolbug.pamphlet @@ -6,7 +6,7 @@ \chapter{Introduction} \section{The Numbering Scheme} \begin{verbatim} -bug 7321: +bug 7322: todo 342: wish 1012: meh 5: @@ -32,6 +32,18 @@ The books/endpaper.pamphlet should be added to the Jenks book \chapter{book5 Interpreter} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{bug 7321: missing spaces in )show output} +\begin{verbatim} +)show INTABL + InnerTable(Key: SetCategory,Entry: SetCategory,addDom)where + addDom: TableAggregate(Key,Entry)with + finiteAggregate is a domain constructor + Abbreviation for InnerTable is INTABL + This constructor is not exposed in this frame. + Issue )edit bookvol10.3.pamphlet to see algebra source code for INTABL + +\end{verbatim} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{bug 7237: coerce failure} \begin{verbatim} @@ -35032,14 +35044,15 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib SUMFS Warnings: - [1] sum: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE sum (F F (Symbol))) (SIGNATURE sum (F F (SegmentBinding F)))) - [2] notRF?: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE sum (F F (Symbol))) (SIGNATURE sum (F F (SegmentBinding F)))) + [1] sum: not known that (OrderedSet) is of mode (CATEGORY package + [2] notRF?: not known that (OrderedSet) is of mode (CATEGORY package (|UnivariateTaylorSeriesCategory| |#1|) extends (|UnivariatePowerSeriesCategory| |#1| (|NonNegativeInteger|)) but not -(|UnivariatePowerSeriesCategory| |#1| (|Integer|)) --------------non extending category---------------------- +(|UnivariatePowerSeriesCategory| |#1| (|Integer|)) --------------non +extending category---------------------- .. SparseUnivariateTaylorSeries(#1,#2,#3) of cat -(|Join| (|UnivariateTaylorSeriesCategory| |#1|) (CATEGORY |domain| (SIGNATURE |coerce| ($ (|UnivariatePolynomial| |#2| |#1|))) (SIGNATURE |univariatePolynomial| ((|UnivariatePolynomial| |#2| |#1|) $ (|NonNegativeInteger|))) (SIGNATURE |coerce| ($ (|Variable| |#2|))) (SIGNATURE |differentiate| ($ $ (|Variable| |#2|))) (IF (|has| |#1| (|Algebra| (|Fraction| (|Integer|)))) (SIGNATURE |integrate| ($ $ (|Variable| |#2|))) |noBranch|))) has no +(|Join| (|UnivariateTaylorSeriesCategory| |#1|) (CATEGORY |domain| \end{verbatim} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -35059,8 +35072,8 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib TOOLSIGN Warnings: [1] nonQsign: pretend(AlgebraicNumber) -- should replace by @ - [2] nonQsign: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has (Integer) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Integer))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Integer) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Integer) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [3] nonQsign: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Integer) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Integer))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Integer) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Integer) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) + [2] nonQsign: not known that (AlgebraicallyClosedField) is of mode + [3] nonQsign: not known that (TranscendentalFunctionCategory) is of \end{verbatim} @@ -35071,15 +35084,15 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib TRIGMNIP Warnings: - [1] K2KG: not known that (RadicalCategory) is of mode (CATEGORY domain (IF (has (Complex R) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Complex R))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Complex R) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Complex R) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [2] K2KG: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Complex R) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Complex R))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Complex R) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Complex R) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [3] K2KG: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE complexNormalize (F F)) (SIGNATURE complexNormalize (F F (Symbol))) (SIGNATURE complexElementary (F F)) (SIGNATURE complexElementary (F F (Symbol))) (SIGNATURE trigs (F F)) (SIGNATURE real (F F)) (SIGNATURE imag (F F)) (SIGNATURE real? ((Boolean) F)) (SIGNATURE complexForm ((Complex F) F))) + [1] K2KG: not known that (RadicalCategory) is of mode (CATEGORY domain + [2] K2KG: not known that (TranscendentalFunctionCategory) is of mode + [3] K2KG: not known that (OrderedSet) is of mode (CATEGORY package [4] real?: ker has no value - [5] complexKernels: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE complexNormalize (F F)) (SIGNATURE complexNormalize (F F (Symbol))) (SIGNATURE complexElementary (F F)) (SIGNATURE complexElementary (F F (Symbol))) (SIGNATURE trigs (F F)) (SIGNATURE real (F F)) (SIGNATURE imag (F F)) (SIGNATURE real? ((Boolean) F)) (SIGNATURE complexForm ((Complex F) F))) + [5] complexKernels: not known that (OrderedSet) is of mode (CATEGORY [6] complexKernels: lk has no value [7] complexKernels: lv has no value - [8] complexNormalize: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has (Complex R) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Complex R))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Complex R) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Complex R) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [9] complexNormalize: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Complex R) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Complex R))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Complex R) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Complex R) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) + [8] complexNormalize: not known that (AlgebraicallyClosedField) is of + [9] complexNormalize: not known that (TranscendentalFunctionCategory) \end{verbatim} @@ -35090,16 +35103,16 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib TRMANIP Warnings: - [1] logArgs: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE expand (F F)) (SIGNATURE simplify (F F)) (SIGNATURE htrigs (F F)) (SIGNATURE simplifyExp (F F)) (SIGNATURE simplifyLog (F F)) (SIGNATURE expandPower (F F)) (SIGNATURE expandLog (F F)) (SIGNATURE cos2sec (F F)) (SIGNATURE cosh2sech (F F)) (SIGNATURE cot2trig (F F)) (SIGNATURE coth2trigh (F F)) (SIGNATURE csc2sin (F F)) (SIGNATURE csch2sinh (F F)) (SIGNATURE sec2cos (F F)) (SIGNATURE sech2cosh (F F)) (SIGNATURE sin2csc (F F)) (SIGNATURE sinh2csch (F F)) (SIGNATURE tan2trig (F F)) (SIGNATURE tanh2trigh (F F)) (SIGNATURE tan2cot (F F)) (SIGNATURE tanh2coth (F F)) (SIGNATURE cot2tan (F F)) (SIGNATURE coth2tanh (F F)) (SIGNATURE removeCosSq (F F)) (SIGNATURE removeSinSq (F F)) (SIGNATURE removeCoshSq (F F)) (SIGNATURE removeSinhSq (F F)) (IF (has R (PatternMatchable R)) (IF (has R (ConvertibleTo (Pattern R))) (IF (has F (ConvertibleTo (Pattern R))) (IF (has F (PatternMatchable R)) (SIGNATURE expandTrigProducts (F F)) noBranch) noBranch) noBranch) noBranch)) + [1] logArgs: not known that (OrderedSet) is of mode (CATEGORY package [2] logArgs: sum has no value [3] logArgs: arg has no value [4] simplifyLog1: exprs has no value [5] simplifyLog1: terms has no value - [6] simplifyLog1: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE expand (F F)) (SIGNATURE simplify (F F)) (SIGNATURE htrigs (F F)) (SIGNATURE simplifyExp (F F)) (SIGNATURE simplifyLog (F F)) (SIGNATURE expandPower (F F)) (SIGNATURE expandLog (F F)) (SIGNATURE cos2sec (F F)) (SIGNATURE cosh2sech (F F)) (SIGNATURE cot2trig (F F)) (SIGNATURE coth2trigh (F F)) (SIGNATURE csc2sin (F F)) (SIGNATURE csch2sinh (F F)) (SIGNATURE sec2cos (F F)) (SIGNATURE sech2cosh (F F)) (SIGNATURE sin2csc (F F)) (SIGNATURE sinh2csch (F F)) (SIGNATURE tan2trig (F F)) (SIGNATURE tanh2trigh (F F)) (SIGNATURE tan2cot (F F)) (SIGNATURE tanh2coth (F F)) (SIGNATURE cot2tan (F F)) (SIGNATURE coth2tanh (F F)) (SIGNATURE removeCosSq (F F)) (SIGNATURE removeSinSq (F F)) (SIGNATURE removeCoshSq (F F)) (SIGNATURE removeSinhSq (F F)) (IF (has R (PatternMatchable R)) (IF (has R (ConvertibleTo (Pattern R))) (IF (has F (ConvertibleTo (Pattern R))) (IF (has F (PatternMatchable R)) (SIGNATURE expandTrigProducts (F F)) noBranch) noBranch) noBranch) noBranch)) - [7] expandpow: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE expand (F F)) (SIGNATURE simplify (F F)) (SIGNATURE htrigs (F F)) (SIGNATURE simplifyExp (F F)) (SIGNATURE simplifyLog (F F)) (SIGNATURE expandPower (F F)) (SIGNATURE expandLog (F F)) (SIGNATURE cos2sec (F F)) (SIGNATURE cosh2sech (F F)) (SIGNATURE cot2trig (F F)) (SIGNATURE coth2trigh (F F)) (SIGNATURE csc2sin (F F)) (SIGNATURE csch2sinh (F F)) (SIGNATURE sec2cos (F F)) (SIGNATURE sech2cosh (F F)) (SIGNATURE sin2csc (F F)) (SIGNATURE sinh2csch (F F)) (SIGNATURE tan2trig (F F)) (SIGNATURE tanh2trigh (F F)) (SIGNATURE tan2cot (F F)) (SIGNATURE tanh2coth (F F)) (SIGNATURE cot2tan (F F)) (SIGNATURE coth2tanh (F F)) (SIGNATURE removeCosSq (F F)) (SIGNATURE removeSinSq (F F)) (SIGNATURE removeCoshSq (F F)) (SIGNATURE removeSinhSq (F F)) (IF (has R (PatternMatchable R)) (IF (has R (ConvertibleTo (Pattern R))) (IF (has F (ConvertibleTo (Pattern R))) (IF (has F (PatternMatchable R)) (SIGNATURE expandTrigProducts (F F)) noBranch) noBranch) noBranch) noBranch)) + [6] simplifyLog1: not known that (OrderedSet) is of mode (CATEGORY + [7] expandpow: not known that (OrderedSet) is of mode (CATEGORY [8] termexp: exponent has no value - [9] htrigs: not known that (Ring) is of mode (CATEGORY package (SIGNATURE expand (F F)) (SIGNATURE simplify (F F)) (SIGNATURE htrigs (F F)) (SIGNATURE simplifyExp (F F)) (SIGNATURE simplifyLog (F F)) (SIGNATURE expandPower (F F)) (SIGNATURE expandLog (F F)) (SIGNATURE cos2sec (F F)) (SIGNATURE cosh2sech (F F)) (SIGNATURE cot2trig (F F)) (SIGNATURE coth2trigh (F F)) (SIGNATURE csc2sin (F F)) (SIGNATURE csch2sinh (F F)) (SIGNATURE sec2cos (F F)) (SIGNATURE sech2cosh (F F)) (SIGNATURE sin2csc (F F)) (SIGNATURE sinh2csch (F F)) (SIGNATURE tan2trig (F F)) (SIGNATURE tanh2trigh (F F)) (SIGNATURE tan2cot (F F)) (SIGNATURE tanh2coth (F F)) (SIGNATURE cot2tan (F F)) (SIGNATURE coth2tanh (F F)) (SIGNATURE removeCosSq (F F)) (SIGNATURE removeSinSq (F F)) (SIGNATURE removeCoshSq (F F)) (SIGNATURE removeSinhSq (F F)) (IF (has R (PatternMatchable R)) (IF (has R (ConvertibleTo (Pattern R))) (IF (has F (ConvertibleTo (Pattern R))) (IF (has F (PatternMatchable R)) (SIGNATURE expandTrigProducts (F F)) noBranch) noBranch) noBranch) noBranch)) - [10] exlog: not known that (IntegralDomain) is of mode (CATEGORY package (SIGNATURE expand (F F)) (SIGNATURE simplify (F F)) (SIGNATURE htrigs (F F)) (SIGNATURE simplifyExp (F F)) (SIGNATURE simplifyLog (F F)) (SIGNATURE expandPower (F F)) (SIGNATURE expandLog (F F)) (SIGNATURE cos2sec (F F)) (SIGNATURE cosh2sech (F F)) (SIGNATURE cot2trig (F F)) (SIGNATURE coth2trigh (F F)) (SIGNATURE csc2sin (F F)) (SIGNATURE csch2sinh (F F)) (SIGNATURE sec2cos (F F)) (SIGNATURE sech2cosh (F F)) (SIGNATURE sin2csc (F F)) (SIGNATURE sinh2csch (F F)) (SIGNATURE tan2trig (F F)) (SIGNATURE tanh2trigh (F F)) (SIGNATURE tan2cot (F F)) (SIGNATURE tanh2coth (F F)) (SIGNATURE cot2tan (F F)) (SIGNATURE coth2tanh (F F)) (SIGNATURE removeCosSq (F F)) (SIGNATURE removeSinSq (F F)) (SIGNATURE removeCoshSq (F F)) (SIGNATURE removeSinhSq (F F)) (IF (has R (PatternMatchable R)) (IF (has R (ConvertibleTo (Pattern R))) (IF (has F (ConvertibleTo (Pattern R))) (IF (has F (PatternMatchable R)) (SIGNATURE expandTrigProducts (F F)) noBranch) noBranch) noBranch) noBranch)) + [9] htrigs: not known that (Ring) is of mode (CATEGORY package + [10] exlog: not known that (IntegralDomain) is of mode [11] logexpand: IN has no value [12] logexpand: x has no value [13] kerexpand: IN has no value @@ -35129,7 +35142,8 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib UPXSSING Warnings: - [1] retractIfCan: signature of lhs not unique: (Union (UnivariatePuiseuxSeries FE var cen) failed)$ chosen + [1] retractIfCan: signature of lhs not unique: (Union +(UnivariatePuiseuxSeries FE var cen) failed)$ chosen [2] sortAndDiscardTerms: zeroTerms has no value [3] sortAndDiscardTerms: infiniteTerms has no value [4] sortAndDiscardTerms: failedTerms has no value @@ -35138,7 +35152,7 @@ Warning: PFO;cmult has a duplicate definition in this file --------------non extending category---------------------- .. UnivariatePuiseuxSeriesWithExponentialSingularity(#1,#2,#3,#4) of cat -(|Join| (|FiniteAbelianMonoidRing| (|UnivariatePuiseuxSeries| |#2| |#3| |#4|) (|ExponentialOfUnivariatePuiseuxSeries| |#2| |#3| |#4|)) (|IntegralDomain|) (CATEGORY |domain| (SIGNATURE |limitPlus| ((|Union| (|OrderedCompletion| |#2|) "failed") $)) (SIGNATURE |dominantTerm| ((|Union| (|Record| (|:| |%term| (|Record| (|:| |%coef| (|UnivariatePuiseuxSeries| |#2| |#3| |#4|)) (|:| |%expon| (|ExponentialOfUnivariatePuiseuxSeries| |#2| |#3| |#4|)) (|:| |%expTerms| (|List| (|Record| (|:| |k| (|Fraction| (|Integer|))) (|:| |c| |#2|)))))) (|:| |%type| (|String|))) "failed") $)))) has no +(|Join| (|FiniteAbelianMonoidRing| (|UnivariatePuiseuxSeries| |#2| |#3| |#4|) \end{verbatim} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -35146,7 +35160,7 @@ Warning: PFO;cmult has a duplicate definition in this file \begin{verbatim} -(IF (|has| (|UnivariatePuiseuxSeries| |#2| |#3| |#4|) (|IntegralDomain|)) (IF (|has| (|ExponentialOfUnivariatePuiseuxSeries| |#2| |#3| |#4|) (|CancellationAbelianMonoid|)) (SIGNATURE |fmecg| ($ $ (|ExponentialOfUnivariatePuiseuxSeries| |#2| |#3| |#4|) (|UnivariatePuiseuxSeries| |#2| |#3| |#4|) $)) |noBranch|) |noBranch|) finalizing nrlib UPXSSING +(IF (|has| (|UnivariatePuiseuxSeries| |#2| |#3| |#4|) (|IntegralDomain|)) \end{verbatim} @@ -35174,9 +35188,9 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib DEFINTEF Warnings: - [1] checkForPole: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE integrate ((Union (: f1 (OrderedCompletion F)) (: f2 (List (OrderedCompletion F))) (: fail failed) (: pole potentialPole)) F (SegmentBinding (OrderedCompletion F)))) (SIGNATURE integrate ((Union (: f1 (OrderedCompletion F)) (: f2 (List (OrderedCompletion F))) (: fail failed) (: pole potentialPole)) F (SegmentBinding (OrderedCompletion F)) (String))) (SIGNATURE innerint ((Union (: f1 (OrderedCompletion F)) (: f2 (List (OrderedCompletion F))) (: fail failed) (: pole potentialPole)) F (Symbol) (OrderedCompletion F) (OrderedCompletion F) (Boolean)))) - [2] polyIfCan: not known that (Ring) is of mode (CATEGORY package (SIGNATURE integrate ((Union (: f1 (OrderedCompletion F)) (: f2 (List (OrderedCompletion F))) (: fail failed) (: pole potentialPole)) F (SegmentBinding (OrderedCompletion F)))) (SIGNATURE integrate ((Union (: f1 (OrderedCompletion F)) (: f2 (List (OrderedCompletion F))) (: fail failed) (: pole potentialPole)) F (SegmentBinding (OrderedCompletion F)) (String))) (SIGNATURE innerint ((Union (: f1 (OrderedCompletion F)) (: f2 (List (OrderedCompletion F))) (: fail failed) (: pole potentialPole)) F (Symbol) (OrderedCompletion F) (OrderedCompletion F) (Boolean)))) - [3] polyIfCan: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE integrate ((Union (: f1 (OrderedCompletion F)) (: f2 (List (OrderedCompletion F))) (: fail failed) (: pole potentialPole)) F (SegmentBinding (OrderedCompletion F)))) (SIGNATURE integrate ((Union (: f1 (OrderedCompletion F)) (: f2 (List (OrderedCompletion F))) (: fail failed) (: pole potentialPole)) F (SegmentBinding (OrderedCompletion F)) (String))) (SIGNATURE innerint ((Union (: f1 (OrderedCompletion F)) (: f2 (List (OrderedCompletion F))) (: fail failed) (: pole potentialPole)) F (Symbol) (OrderedCompletion F) (OrderedCompletion F) (Boolean)))) + [1] checkForPole: not known that (OrderedSet) is of mode (CATEGORY + [2] polyIfCan: not known that (Ring) is of mode (CATEGORY package + [3] polyIfCan: not known that (OrderedSet) is of mode (CATEGORY package \end{verbatim} @@ -35187,9 +35201,9 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib DFINTTLS Warnings: - [1] findLimit: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE ignore? ((Boolean) (String))) (SIGNATURE computeInt ((Union (OrderedCompletion F) failed) (Kernel F) F (OrderedCompletion F) (OrderedCompletion F) (Boolean))) (SIGNATURE checkForZero ((Union (Boolean) failed) (Polynomial R) (Symbol) (OrderedCompletion F) (OrderedCompletion F) (Boolean))) (SIGNATURE checkForZero ((Union (Boolean) failed) (SparseUnivariatePolynomial F) (OrderedCompletion F) (OrderedCompletion F) (Boolean)))) - [2] mkLogPos: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE ignore? ((Boolean) (String))) (SIGNATURE computeInt ((Union (OrderedCompletion F) failed) (Kernel F) F (OrderedCompletion F) (OrderedCompletion F) (Boolean))) (SIGNATURE checkForZero ((Union (Boolean) failed) (Polynomial R) (Symbol) (OrderedCompletion F) (OrderedCompletion F) (Boolean))) (SIGNATURE checkForZero ((Union (Boolean) failed) (SparseUnivariatePolynomial F) (OrderedCompletion F) (OrderedCompletion F) (Boolean)))) - [3] checkForZero: not known that (Ring) is of mode (CATEGORY package (SIGNATURE ignore? ((Boolean) (String))) (SIGNATURE computeInt ((Union (OrderedCompletion F) failed) (Kernel F) F (OrderedCompletion F) (OrderedCompletion F) (Boolean))) (SIGNATURE checkForZero ((Union (Boolean) failed) (Polynomial R) (Symbol) (OrderedCompletion F) (OrderedCompletion F) (Boolean))) (SIGNATURE checkForZero ((Union (Boolean) failed) (SparseUnivariatePolynomial F) (OrderedCompletion F) (OrderedCompletion F) (Boolean)))) + [1] findLimit: not known that (OrderedSet) is of mode (CATEGORY package + [2] mkLogPos: not known that (OrderedSet) is of mode (CATEGORY package + [3] checkForZero: not known that (Ring) is of mode (CATEGORY package [4] findRealZero: fin has no value [5] findRealZero: halfinf has no value [6] var: i has no value @@ -35203,8 +35217,8 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib DEFINTRF Warnings: - [1] nopole: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has R (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has R (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [2] nopole: not known that (AlgebraicallyClosedFunctionSpace R) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has R (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has R (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) + [1] nopole: not known that (TranscendentalFunctionCategory) is of mode + [2] nopole: not known that (AlgebraicallyClosedFunctionSpace R) is of \end{verbatim} @@ -35215,7 +35229,7 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib D01TRNS Warnings: - [1] transformFunction: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Fraction (Integer)) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Fraction (Integer)))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Fraction (Integer)) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Fraction (Integer)) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) + [1] transformFunction: not known that (TranscendentalFunctionCategory) --------------non extending category---------------------- .. d01TransformFunctionType of cat @@ -35237,7 +35251,8 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib D01TRNS ---->/research2/test0819/mnt/fedora5/../../src/algebra/D01TRNS.spad-->d01TransformFunctionType(): Missing Description +--->/research2/test0819/mnt/fedora5/../../src/algebra/D01TRNS.spad +-->d01TransformFunctionType(): Missing Description \end{verbatim} @@ -35248,7 +35263,7 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib EFULS Warnings: - [1] tanIfCan: not known that (Algebra (Fraction (Integer))) is of mode (CATEGORY Coef (SIGNATURE retractIfCan ((Union (Fraction (Integer)) failed) Coef))) + [1] tanIfCan: not known that (Algebra (Fraction (Integer))) is \end{verbatim} @@ -35259,7 +35274,7 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib ESCONT Warnings: - [1] zerosOf: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE getlo ((DoubleFloat) (Segment (OrderedCompletion (DoubleFloat))))) (SIGNATURE gethi ((DoubleFloat) (Segment (OrderedCompletion (DoubleFloat))))) (SIGNATURE functionIsFracPolynomial? ((Boolean) (Record (: var (Symbol)) (: fn (Expression (DoubleFloat))) (: range (Segment (OrderedCompletion (DoubleFloat)))) (: abserr (DoubleFloat)) (: relerr (DoubleFloat))))) (SIGNATURE problemPoints ((List (DoubleFloat)) (Expression (DoubleFloat)) (Symbol) (Segment (OrderedCompletion (DoubleFloat))))) (SIGNATURE zerosOf ((Stream (DoubleFloat)) (Expression (DoubleFloat)) (List (Symbol)) (Segment (OrderedCompletion (DoubleFloat))))) (SIGNATURE singularitiesOf ((Stream (DoubleFloat)) (Expression (DoubleFloat)) (List (Symbol)) (Segment (OrderedCompletion (DoubleFloat))))) (SIGNATURE singularitiesOf ((Stream (DoubleFloat)) (Vector (Expression (DoubleFloat))) (List (Symbol)) (Segment (OrderedCompletion (DoubleFloat))))) (SIGNATURE polynomialZeros ((List (DoubleFloat)) (Polynomial (Fraction (Integer))) (Symbol) (Segment (OrderedCompletion (DoubleFloat))))) (SIGNATURE df2st ((String) (DoubleFloat))) (SIGNATURE ldf2lst ((List (String)) (List (DoubleFloat)))) (SIGNATURE sdf2lst ((List (String)) (Stream (DoubleFloat))))) + [1] zerosOf: not known that (OrderedSet) is of mode (CATEGORY \end{verbatim} @@ -35271,7 +35286,7 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib ESCONT --------------non extending category---------------------- .. ExpertSystemContinuityPackage of cat -(CATEGORY |package| (SIGNATURE |getlo| ((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (SIGNATURE |gethi| ((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (SIGNATURE |functionIsFracPolynomial?| ((|Boolean|) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|))))) (SIGNATURE |problemPoints| ((|List| (|DoubleFloat|)) (|Expression| (|DoubleFloat|)) (|Symbol|) (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (SIGNATURE |zerosOf| ((|Stream| (|DoubleFloat|)) (|Expression| (|DoubleFloat|)) (|List| (|Symbol|)) (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (SIGNATURE |singularitiesOf| ((|Stream| (|DoubleFloat|)) (|Expression| (|DoubleFloat|)) (|List| (|Symbol|)) (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (SIGNATURE |singularitiesOf| ((|Stream| (|DoubleFloat|)) (|Vector| (|Expression| (|DoubleFloat|))) (|List| (|Symbol|)) (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (SIGNATURE |polynomialZeros| ((|List| (|DoubleFloat|)) (|Polynomial| (|Fraction| (|Integer|))) (|Symbol|) (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (SIGNATURE |df2st| ((|String|) (|DoubleFloat|))) (SIGNATURE |ldf2lst| ((|List| (|String|)) (|List| (|DoubleFloat|)))) (SIGNATURE |sdf2lst| ((|List| (|String|)) (|Stream| (|DoubleFloat|))))) has no f2df : Float -> DoubleFloat +(CATEGORY |package| (SIGNATURE |getlo| ((|DoubleFloat|) (|Segment| \end{verbatim} @@ -35282,55 +35297,55 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib EXPR Warnings: - [1] not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [2] simplifyPower: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) + [1] not known that (OrderedSet) is of mode (CATEGORY domain + [2] simplifyPower: not known that (OrderedSet) is of mode [3] **: pretend(Integer) -- should replace by @ - [4] **: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [5] <: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [6] numer: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [7] toprat: not known that (Field) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [8] toprat: not known that (ExpressionSpace) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [9] toprat: not known that (SIGNATURE numer ((SparseMultivariatePolynomial R (Kernel $)) $)) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [10] toprat: not known that (SIGNATURE denom ((SparseMultivariatePolynomial R (Kernel $)) $)) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [11] toprat: not known that (SIGNATURE coerce ($ (SparseMultivariatePolynomial R (Kernel $)))) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [12] reducedSystem: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [13] commonk0: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [14] rootOf: not known that (FunctionSpace R) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [15] rootOf: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [16] pi: not known that (FunctionSpace R) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [17] pi: not known that (RadicalCategory) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [18] abs: not known that (FunctionSpace R) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [19] **: not known that (FunctionSpace R) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [20] erf: not known that (FunctionSpace R) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [21] erf: not known that (RadicalCategory) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [22] erf: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [23] operator: not known that (FunctionSpace R) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [24] operator: not known that (ExpressionSpace) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [25] evl0: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [26] gcdPolynomial: not known that (GcdDomain) is of mode (CATEGORY domain (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) - [27] factorPolynomial: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) - [28] factorPolynomial: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) - [29] coerce: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [30] retract: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [31] retractIfCan: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [32] k2expr: not known that (ExpressionSpace) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [33] smp2expr: not known that (SetCategory) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [34] smp2expr: not known that (SIGNATURE + ($ $ $)) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [35] smp2expr: not known that (SIGNATURE * ($ $ $)) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [36] smp2expr: not known that (SIGNATURE ** ($ $ (NonNegativeInteger))) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [37] smp2an: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [38] convert: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [39] eval: not known that (ConvertibleTo (InputForm)) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [40] patternMatch: not known that (FunctionSpace R) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [41] patternMatch: not known that (ConvertibleTo (Pattern (Integer))) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [42] patternMatch: not known that (PatternMatchable (Integer)) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [43] patternMatch: not known that (RetractableTo (Kernel $)) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [44] patternMatch: not known that (SetCategory) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [45] patternMatch: not known that (ConvertibleTo (Pattern (Float))) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [46] patternMatch: not known that (PatternMatchable (Float)) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) + [4] **: not known that (OrderedSet) is of mode (CATEGORY domain + [5] <: not known that (OrderedSet) is of mode (CATEGORY domain + [6] numer: not known that (OrderedSet) is of mode (CATEGORY domain + [7] toprat: not known that (Field) is of mode (CATEGORY domain + [8] toprat: not known that (ExpressionSpace) is of mode + [9] toprat: not known that (SIGNATURE numer + [10] toprat: not known that (SIGNATURE denom + [11] toprat: not known that (SIGNATURE coerce ($ + [12] reducedSystem: not known that (Ring) is of mode (CATEGORY domain + [13] commonk0: not known that (OrderedSet) is of mode (CATEGORY domain + [14] rootOf: not known that (FunctionSpace R) is of mode (CATEGORY + [15] rootOf: not known that (Ring) is of mode (CATEGORY domain + [16] pi: not known that (FunctionSpace R) is of mode (CATEGORY domain + [17] pi: not known that (RadicalCategory) is of mode (CATEGORY domain + [18] abs: not known that (FunctionSpace R) is of mode (CATEGORY domain + [19] **: not known that (FunctionSpace R) is of mode (CATEGORY domain + [20] erf: not known that (FunctionSpace R) is of mode (CATEGORY domain + [21] erf: not known that (RadicalCategory) is of mode (CATEGORY domain + [22] erf: not known that (TranscendentalFunctionCategory) is of mode + [23] operator: not known that (FunctionSpace R) is of mode (CATEGORY + [24] operator: not known that (ExpressionSpace) is of mode (CATEGORY + [25] evl0: not known that (OrderedSet) is of mode (CATEGORY domain + [26] gcdPolynomial: not known that (GcdDomain) is of mode (CATEGORY + [27] factorPolynomial: not known that (OrderedSet) is of mode + [28] factorPolynomial: not known that (Ring) is of mode (CATEGORY + [29] coerce: not known that (OrderedSet) is of mode (CATEGORY domain + [30] retract: not known that (OrderedSet) is of mode (CATEGORY domain + [31] retractIfCan: not known that (OrderedSet) is of mode (CATEGORY + [32] k2expr: not known that (ExpressionSpace) is of mode (CATEGORY + [33] smp2expr: not known that (SetCategory) is of mode (CATEGORY + [34] smp2expr: not known that (SIGNATURE + ($ $ $)) is of mode + [35] smp2expr: not known that (SIGNATURE * ($ $ $)) is of mode + [36] smp2expr: not known that (SIGNATURE ** ($ $ (NonNegativeInteger))) + [37] smp2an: not known that (OrderedSet) is of mode (CATEGORY domain + [38] convert: not known that (OrderedSet) is of mode (CATEGORY domain + [39] eval: not known that (ConvertibleTo (InputForm)) is of mode + [40] patternMatch: not known that (FunctionSpace R) is of mode + [41] patternMatch: not known that (ConvertibleTo (Pattern (Integer))) + [42] patternMatch: not known that (PatternMatchable (Integer)) is of + [43] patternMatch: not known that (RetractableTo (Kernel $)) is of + [44] patternMatch: not known that (SetCategory) is of mode (CATEGORY + [45] patternMatch: not known that (ConvertibleTo (Pattern (Float))) + [46] patternMatch: not known that (PatternMatchable (Float)) is of [47] isPlus: gen has no value - [48] not known that (Ring) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) - [49] not known that (IntegralDomain) is of mode (CATEGORY domain (SIGNATURE simplifyPower ($ $ (Integer)))) + [48] not known that (Ring) is of mode (CATEGORY domain (SIGNATURE + [49] not known that (IntegralDomain) is of mode (CATEGORY domain \end{verbatim} @@ -35355,16 +35370,24 @@ Warning: PFO;cmult has a duplicate definition in this file Warnings: [1] iTaylor: %problem has no value [2] iTaylor: %series has no value - [3] taylor: more than 1 modemap for: (Zero) with dc=FE ===>(((FE FE) ((has R (AbelianSemiGroup)) (CONST FE ($)))) ((FE FE) (T (CONST FE ($))))) + [3] taylor: more than 1 modemap for: (Zero) with dc=FE ===> +(((FE FE) ((has R (AbelianSemiGroup)) (CONST FE ($)))) ((FE FE) +(T (CONST FE ($))))) [4] iLaurent: %problem has no value [5] iLaurent: %series has no value - [6] laurent: more than 1 modemap for: (Zero) with dc=FE ===>(((FE FE) ((has R (AbelianSemiGroup)) (CONST FE ($)))) ((FE FE) (T (CONST FE ($))))) + [6] laurent: more than 1 modemap for: (Zero) with dc=FE ===> +(((FE FE) ((has R (AbelianSemiGroup)) (CONST FE ($)))) ((FE FE) +(T (CONST FE ($))))) [7] iPuiseux: %problem has no value [8] iPuiseux: %series has no value - [9] puiseux: more than 1 modemap for: (Zero) with dc=FE ===>(((FE FE) ((has R (AbelianSemiGroup)) (CONST FE ($)))) ((FE FE) (T (CONST FE ($))))) + [9] puiseux: more than 1 modemap for: (Zero) with dc=FE ===> +(((FE FE) ((has R (AbelianSemiGroup)) (CONST FE ($)))) ((FE FE) +(T (CONST FE ($))))) [10] iSeries: %problem has no value [11] iSeries: %series has no value - [12] series: more than 1 modemap for: (Zero) with dc=FE ===>(((FE FE) ((has R (AbelianSemiGroup)) (CONST FE ($)))) ((FE FE) (T (CONST FE ($))))) + [12] series: more than 1 modemap for: (Zero) with dc=FE ===> +(((FE FE) ((has R (AbelianSemiGroup)) (CONST FE ($)))) ((FE FE) +(T (CONST FE ($))))) \end{verbatim} @@ -35398,12 +35421,12 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib FSCINT Warnings: - [1] K2KG: not known that (RadicalCategory) is of mode (CATEGORY domain (IF (has (Complex R) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Complex R))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Complex R) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Complex R) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [2] K2KG: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Complex R) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Complex R))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Complex R) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Complex R) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [3] K2KG: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE internalIntegrate ((IntegrationResult F) F (Symbol))) (SIGNATURE internalIntegrate0 ((IntegrationResult F) F (Symbol))) (SIGNATURE complexIntegrate (F F (Symbol)))) - [4] internalIntegrate: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE internalIntegrate ((IntegrationResult F) F (Symbol))) (SIGNATURE internalIntegrate0 ((IntegrationResult F) F (Symbol))) (SIGNATURE complexIntegrate (F F (Symbol)))) - [5] internalIntegrate: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has (Complex R) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Complex R))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Complex R) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Complex R) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [6] internalIntegrate: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Complex R) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Complex R))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Complex R) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Complex R) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) + [1] K2KG: not known that (RadicalCategory) is of mode (CATEGORY domain + [2] K2KG: not known that (TranscendentalFunctionCategory) is of mode + [3] K2KG: not known that (OrderedSet) is of mode (CATEGORY package + [4] internalIntegrate: not known that (OrderedSet) is of mode (CATEGORY + [5] internalIntegrate: not known that (AlgebraicallyClosedField) is of + [6] internalIntegrate: not known that (TranscendentalFunctionCategory) \end{verbatim} @@ -35414,13 +35437,13 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib FSINT Warnings: - [1] K2KG: not known that (RadicalCategory) is of mode (CATEGORY domain (IF (has (Complex R) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Complex R))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Complex R) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Complex R) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [2] K2KG: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Complex R) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Complex R))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Complex R) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Complex R) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [3] K2KG: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE integrate ((Union F (List F)) F (Symbol)))) - [4] postSubst: not known that (Ring) is of mode (CATEGORY package (SIGNATURE integrate ((Union F (List F)) F (Symbol)))) - [5] postSubst: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE integrate ((Union F (List F)) F (Symbol)))) - [6] integrate: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has (Complex R) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Complex R))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Complex R) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Complex R) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [7] integrate: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Complex R) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Complex R))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Complex R) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Complex R) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) + [1] K2KG: not known that (RadicalCategory) is of mode (CATEGORY domain + [2] K2KG: not known that (TranscendentalFunctionCategory) is of mode + [3] K2KG: not known that (OrderedSet) is of mode (CATEGORY package + [4] postSubst: not known that (Ring) is of mode (CATEGORY package + [5] postSubst: not known that (OrderedSet) is of mode (CATEGORY + [6] integrate: not known that (AlgebraicallyClosedField) is of mode + [7] integrate: not known that (TranscendentalFunctionCategory) is of \end{verbatim} @@ -35431,8 +35454,8 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib FS2EXPXP Warnings: - [1] newElem: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE exprToXXP ((Union (: %expansion (ExponentialExpansion R FE x cen)) (: %problem (Record (: func (String)) (: prob (String))))) FE (Boolean))) (SIGNATURE localAbs (FE FE))) - [2] k2Elem: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE exprToXXP ((Union (: %expansion (ExponentialExpansion R FE x cen)) (: %problem (Record (: func (String)) (: prob (String))))) FE (Boolean))) (SIGNATURE localAbs (FE FE))) + [1] newElem: not known that (OrderedSet) is of mode + [2] k2Elem: not known that (OrderedSet) is of mode [3] iExprToXXP: %series has no value [4] listToXXP: %expansion has no value [5] powerToXXP: %expansion has no value @@ -35451,13 +35474,13 @@ Warning: PFO;cmult has a duplicate definition in this file [18] applyIfCan: %expansion has no value [19] applyBddIfCan: %problem has no value [20] applyBddIfCan: %expansion has no value - [21] opsInvolvingX: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE exprToXXP ((Union (: %expansion (ExponentialExpansion R FE x cen)) (: %problem (Record (: func (String)) (: prob (String))))) FE (Boolean))) (SIGNATURE localAbs (FE FE))) + [21] opsInvolvingX: not known that (OrderedSet) is of mode [22] atancotToXXP: %problem has no value [23] atancotToXXP: %series has no value --------------non extending category---------------------- .. GeneralUnivariatePowerSeries(#1,#2,#3) of cat -(|Join| (|UnivariatePuiseuxSeriesCategory| |#1|) (CATEGORY |domain| (SIGNATURE |coerce| ($ (|Variable| |#2|))) (SIGNATURE |coerce| ($ (|UnivariatePuiseuxSeries| |#1| |#2| |#3|))) (SIGNATURE |differentiate| ($ $ (|Variable| |#2|))) (IF (|has| |#1| (|Algebra| (|Fraction| (|Integer|)))) (SIGNATURE |integrate| ($ $ (|Variable| |#2|))) |noBranch|))) has no +(|Join| (|UnivariatePuiseuxSeriesCategory| |#1|) (CATEGORY |domain| \end{verbatim} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -35465,7 +35488,7 @@ Warning: PFO;cmult has a duplicate definition in this file \begin{verbatim} -(|UnivariatePuiseuxSeriesConstructorCategory| |#1| (|UnivariateLaurentSeries| |#1| |#2| |#3|)) finalizing nrlib GSERIES +(|UnivariatePuiseuxSeriesConstructorCategory| |#1| \end{verbatim} @@ -35476,7 +35499,8 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib HELLFDIV Warnings: - [1] unknown Functor code (error HyperellipticFiniteDivisor: curve must be hyperelliptic) + [1] unknown Functor code (error HyperellipticFiniteDivisor: +curve must be hyperelliptic) \end{verbatim} @@ -35487,10 +35511,10 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib INVLAPLA Warnings: - [1] ilt: not known that (Ring) is of mode (CATEGORY package (SIGNATURE inverseLaplace ((Union F failed) F (Symbol) (Symbol)))) - [2] ilt: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE inverseLaplace ((Union F failed) F (Symbol) (Symbol)))) - [3] iltsqfr: not known that (IntegralDomain) is of mode (CATEGORY package (SIGNATURE inverseLaplace ((Union F failed) F (Symbol) (Symbol)))) - [4] iltirred: not known that (Ring) is of mode (CATEGORY package (SIGNATURE inverseLaplace ((Union F failed) F (Symbol) (Symbol)))) + [1] ilt: not known that (Ring) is of mode (CATEGORY package + [2] ilt: not known that (OrderedSet) is of mode (CATEGORY package + [3] iltsqfr: not known that (IntegralDomain) is of mode (CATEGORY + [4] iltirred: not known that (Ring) is of mode (CATEGORY package \end{verbatim} @@ -35501,10 +35525,10 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib IR2F Warnings: - [1] evenRoots: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE split ((IntegrationResult F) (IntegrationResult F))) (SIGNATURE expand ((List F) (IntegrationResult F))) (SIGNATURE complexExpand (F (IntegrationResult F)))) - [2] ilog: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE split ((IntegrationResult F) (IntegrationResult F))) (SIGNATURE expand ((List F) (IntegrationResult F))) (SIGNATURE complexExpand (F (IntegrationResult F)))) - [3] ilog: not known that (Ring) is of mode (CATEGORY package (SIGNATURE split ((IntegrationResult F) (IntegrationResult F))) (SIGNATURE expand ((List F) (IntegrationResult F))) (SIGNATURE complexExpand (F (IntegrationResult F)))) - [4] lg2func: not known that (Ring) is of mode (CATEGORY package (SIGNATURE split ((IntegrationResult F) (IntegrationResult F))) (SIGNATURE expand ((List F) (IntegrationResult F))) (SIGNATURE complexExpand (F (IntegrationResult F)))) + [1] evenRoots: not known that (OrderedSet) is of mode + [2] ilog: not known that (OrderedSet) is of mode (CATEGORY package + [3] ilog: not known that (Ring) is of mode (CATEGORY package + [4] lg2func: not known that (Ring) is of mode (CATEGORY package \end{verbatim} @@ -35515,10 +35539,10 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib IRRF2F Warnings: - [1] expand: not known that (AlgebraicallyClosedFunctionSpace R) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has R (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has R (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [2] expand: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has R (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has R (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [3] integrate: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has R (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has R (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [4] integrate: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has R (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has R (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) + [1] expand: not known that (AlgebraicallyClosedFunctionSpace R) + [2] expand: not known that (TranscendentalFunctionCategory) is of + [3] integrate: not known that (AlgebraicallyClosedField) is of + [4] integrate: not known that (TranscendentalFunctionCategory) \end{verbatim} @@ -35529,11 +35553,11 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib LAPLACE Warnings: - [1] algebraic?: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE laplace (F F (Symbol) (Symbol)))) - [2] isLinear: not known that (Ring) is of mode (CATEGORY package (SIGNATURE laplace (F F (Symbol) (Symbol)))) - [3] isLinear: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE laplace (F F (Symbol) (Symbol)))) + [1] algebraic?: not known that (OrderedSet) is of mode + [2] isLinear: not known that (Ring) is of mode + [3] isLinear: not known that (OrderedSet) is of mode [4] atn: d has no value - [5] mkPlus: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE laplace (F F (Symbol) (Symbol)))) + [5] mkPlus: not known that (OrderedSet) is of mode [6] locallaplace: const has no value [7] locallaplace: nconst has no value @@ -35546,7 +35570,7 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib LIMITPS Warnings: - [1] firstNonLogPtr: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE limit ((Union (OrderedCompletion FE) (Record (: leftHandLimit (Union (OrderedCompletion FE) failed)) (: rightHandLimit (Union (OrderedCompletion FE) failed))) failed) FE (Equation (OrderedCompletion FE)))) (SIGNATURE complexLimit ((Union (OnePointCompletion FE) failed) FE (Equation (OnePointCompletion FE)))) (SIGNATURE limit ((Union (OrderedCompletion FE) failed) FE (Equation FE) (String)))) + [1] firstNonLogPtr: not known that (OrderedSet) is of mode [2] complLimit: %series has no value [3] realLimit: %problem has no value [4] realLimit: %series has no value @@ -35567,10 +35591,10 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib LODEEF Warnings: - [1] algSolve: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE solve ((Union (Record (: particular F) (: basis (List F))) failed) L F (Symbol))) (SIGNATURE solve ((Union F failed) L F (Symbol) F (List F)))) - [2] algSolve: not known that (Ring) is of mode (CATEGORY package (SIGNATURE solve ((Union (Record (: particular F) (: basis (List F))) failed) L F (Symbol))) (SIGNATURE solve ((Union F failed) L F (Symbol) F (List F)))) - [3] xpart: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE solve ((Union (Record (: particular F) (: basis (List F))) failed) L F (Symbol))) (SIGNATURE solve ((Union F failed) L F (Symbol) F (List F)))) - [4] ulodo: not known that (Ring) is of mode (CATEGORY package (SIGNATURE solve ((Union (Record (: particular F) (: basis (List F))) failed) L F (Symbol))) (SIGNATURE solve ((Union F failed) L F (Symbol) F (List F)))) + [1] algSolve: not known that (OrderedSet) is of mode (CATEGORY + [2] algSolve: not known that (Ring) is of mode (CATEGORY package + [3] xpart: not known that (OrderedSet) is of mode (CATEGORY + [4] ulodo: not known that (Ring) is of mode (CATEGORY package \end{verbatim} @@ -35581,8 +35605,8 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib NODE1 Warnings: - [1] solve: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE solve ((Union F failed) F F (BasicOperator) (Symbol)))) - [2] checkBernoulli: not known that (Ring) is of mode (CATEGORY package (SIGNATURE solve ((Union F failed) F F (BasicOperator) (Symbol)))) + [1] solve: not known that (OrderedSet) is of mode (CATEGORY package + [2] checkBernoulli: not known that (Ring) is of mode (CATEGORY package \end{verbatim} @@ -35593,7 +35617,7 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib ODECONST Warnings: - [1] basisSqfr: not known that (Ring) is of mode (CATEGORY package (SIGNATURE constDsolve ((Record (: particular F) (: basis (List F))) L F (Symbol)))) + [1] basisSqfr: not known that (Ring) is of mode \end{verbatim} @@ -35604,10 +35628,10 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib ODEINT Warnings: - [1] expint: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE int (F F (Symbol))) (SIGNATURE expint (F F (Symbol))) (SIGNATURE diff ((Mapping F F) (Symbol)))) + [1] expint: not known that (OrderedSet) is of mode [2] expint: lrec has no value [3] expint: exponent has no value - [4] isQlog: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE int (F F (Symbol))) (SIGNATURE expint (F F (Symbol))) (SIGNATURE diff ((Mapping F F) (Symbol)))) + [4] isQlog: not known that (OrderedSet) is of mode \end{verbatim} @@ -35618,10 +35642,11 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib REP Warnings: - [1] evalvect: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE radicalEigenvectors ((List (Record (: radval (Expression (Integer))) (: radmult (Integer)) (: radvect (List (Matrix (Expression (Integer))))))) (Matrix (Fraction (Polynomial (Integer)))))) (SIGNATURE radicalEigenvector ((List (Matrix (Expression (Integer)))) (Expression (Integer)) (Matrix (Fraction (Polynomial (Integer)))))) (SIGNATURE radicalEigenvalues ((List (Expression (Integer))) (Matrix (Fraction (Polynomial (Integer)))))) (SIGNATURE eigenMatrix ((Union (Matrix (Expression (Integer))) failed) (Matrix (Fraction (Polynomial (Integer)))))) (SIGNATURE normalise ((Matrix (Expression (Integer))) (Matrix (Expression (Integer))))) (SIGNATURE gramschmidt ((List (Matrix (Expression (Integer)))) (List (Matrix (Expression (Integer)))))) (SIGNATURE orthonormalBasis ((List (Matrix (Expression (Integer)))) (Matrix (Fraction (Polynomial (Integer))))))) + [1] evalvect: not known that (OrderedSet) is of mode (CATEGORY [2] gramschmidt: :(PositiveInteger) -- should replace by pretend [3] gramschmidt: :RMR -- should replace by pretend - [4] gramschmidt: :(Matrix (Expression (Integer))) -- should replace by pretend + [4] gramschmidt: :(Matrix (Expression (Integer))) -- +should replace by pretend \end{verbatim} @@ -35647,7 +35672,7 @@ Warning: PFO;cmult has a duplicate definition in this file --------------non extending category---------------------- .. UnivariateTaylorSeries(#1,#2,#3) of cat -(|Join| (|UnivariateTaylorSeriesCategory| |#1|) (CATEGORY |domain| (SIGNATURE |coerce| ($ (|UnivariatePolynomial| |#2| |#1|))) (SIGNATURE |univariatePolynomial| ((|UnivariatePolynomial| |#2| |#1|) $ (|NonNegativeInteger|))) (SIGNATURE |coerce| ($ (|Variable| |#2|))) (SIGNATURE |differentiate| ($ $ (|Variable| |#2|))) (SIGNATURE |lagrange| ($ $)) (SIGNATURE |lambert| ($ $)) (SIGNATURE |oddlambert| ($ $)) (SIGNATURE |evenlambert| ($ $)) (SIGNATURE |generalLambert| ($ $ (|Integer|) (|Integer|))) (SIGNATURE |revert| ($ $)) (SIGNATURE |multisect| ($ (|Integer|) (|Integer|) $)) (SIGNATURE |invmultisect| ($ (|Integer|) (|Integer|) $)) (IF (|has| |#1| (|Algebra| (|Fraction| (|Integer|)))) (SIGNATURE |integrate| ($ $ (|Variable| |#2|))) |noBranch|))) has no ?*? : (%,Integer) -> % +(|Join| (|UnivariateTaylorSeriesCategory| |#1|) (CATEGORY |domain| \end{verbatim} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -35683,9 +35708,9 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib COMBF Warnings: - [1] **: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE belong? ((Boolean) (BasicOperator))) (SIGNATURE operator ((BasicOperator) (BasicOperator))) (SIGNATURE ** (F F F)) (SIGNATURE binomial (F F F)) (SIGNATURE permutation (F F F)) (SIGNATURE factorial (F F)) (SIGNATURE factorials (F F)) (SIGNATURE factorials (F F (Symbol))) (SIGNATURE summation (F F (Symbol))) (SIGNATURE summation (F F (SegmentBinding F))) (SIGNATURE product (F F (Symbol))) (SIGNATURE product (F F (SegmentBinding F))) (SIGNATURE iifact (F F)) (SIGNATURE iibinom (F (List F))) (SIGNATURE iiperm (F (List F))) (SIGNATURE iipow (F (List F))) (SIGNATURE iidsum (F (List F))) (SIGNATURE iidprod (F (List F))) (SIGNATURE ipow (F (List F)))) - [2] facts: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE belong? ((Boolean) (BasicOperator))) (SIGNATURE operator ((BasicOperator) (BasicOperator))) (SIGNATURE ** (F F F)) (SIGNATURE binomial (F F F)) (SIGNATURE permutation (F F F)) (SIGNATURE factorial (F F)) (SIGNATURE factorials (F F)) (SIGNATURE factorials (F F (Symbol))) (SIGNATURE summation (F F (Symbol))) (SIGNATURE summation (F F (SegmentBinding F))) (SIGNATURE product (F F (Symbol))) (SIGNATURE product (F F (SegmentBinding F))) (SIGNATURE iifact (F F)) (SIGNATURE iibinom (F (List F))) (SIGNATURE iiperm (F (List F))) (SIGNATURE iipow (F (List F))) (SIGNATURE iidsum (F (List F))) (SIGNATURE iidprod (F (List F))) (SIGNATURE ipow (F (List F)))) - [3] summand: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE belong? ((Boolean) (BasicOperator))) (SIGNATURE operator ((BasicOperator) (BasicOperator))) (SIGNATURE ** (F F F)) (SIGNATURE binomial (F F F)) (SIGNATURE permutation (F F F)) (SIGNATURE factorial (F F)) (SIGNATURE factorials (F F)) (SIGNATURE factorials (F F (Symbol))) (SIGNATURE summation (F F (Symbol))) (SIGNATURE summation (F F (SegmentBinding F))) (SIGNATURE product (F F (Symbol))) (SIGNATURE product (F F (SegmentBinding F))) (SIGNATURE iifact (F F)) (SIGNATURE iibinom (F (List F))) (SIGNATURE iiperm (F (List F))) (SIGNATURE iipow (F (List F))) (SIGNATURE iidsum (F (List F))) (SIGNATURE iidprod (F (List F))) (SIGNATURE ipow (F (List F)))) + [1] **: not known that (OrderedSet) is of mode (CATEGORY package + [2] facts: not known that (OrderedSet) is of mode (CATEGORY package + [3] summand: not known that (OrderedSet) is of mode (CATEGORY package [4] ipow: n has no value \end{verbatim} @@ -35697,14 +35722,14 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib D01AGNT Warnings: - [1] continuousAtPoint?: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has (Fraction (Integer)) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Fraction (Integer)))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Fraction (Integer)) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Fraction (Integer)) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [2] continuousAtPoint?: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Fraction (Integer)) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Fraction (Integer)))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has (Fraction (Integer)) (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has (Fraction (Integer)) (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [3] functionIsOscillatory: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE rangeIsFinite ((Union (: finite The range is finite) (: lowerInfinite The bottom of range is infinite) (: upperInfinite The top of range is infinite) (: bothInfinite Both top and bottom points are infinite) (: notEvaluated Range not yet evaluated)) (Record (: var (Symbol)) (: fn (Expression (DoubleFloat))) (: range (Segment (OrderedCompletion (DoubleFloat)))) (: abserr (DoubleFloat)) (: relerr (DoubleFloat))))) (SIGNATURE functionIsContinuousAtEndPoints ((Union (: continuous Continuous at the end points) (: lowerSingular There is a singularity at the lower end point) (: upperSingular There is a singularity at the upper end point) (: bothSingular There are singularities at both end points) (: notEvaluated End point continuity not yet evaluated)) (Record (: var (Symbol)) (: fn (Expression (DoubleFloat))) (: range (Segment (OrderedCompletion (DoubleFloat)))) (: abserr (DoubleFloat)) (: relerr (DoubleFloat))))) (SIGNATURE getlo ((DoubleFloat) (Segment (OrderedCompletion (DoubleFloat))))) (SIGNATURE gethi ((DoubleFloat) (Segment (OrderedCompletion (DoubleFloat))))) (SIGNATURE functionIsOscillatory ((Float) (Record (: var (Symbol)) (: fn (Expression (DoubleFloat))) (: range (Segment (OrderedCompletion (DoubleFloat)))) (: abserr (DoubleFloat)) (: relerr (DoubleFloat))))) (SIGNATURE problemPoints ((List (DoubleFloat)) (Expression (DoubleFloat)) (Symbol) (Segment (OrderedCompletion (DoubleFloat))))) (SIGNATURE singularitiesOf ((Stream (DoubleFloat)) (Record (: var (Symbol)) (: fn (Expression (DoubleFloat))) (: range (Segment (OrderedCompletion (DoubleFloat)))) (: abserr (DoubleFloat)) (: relerr (DoubleFloat))))) (SIGNATURE df2st ((String) (DoubleFloat))) (SIGNATURE ldf2lst ((List (String)) (List (DoubleFloat)))) (SIGNATURE sdf2lst ((List (String)) (Stream (DoubleFloat)))) (SIGNATURE commaSeparate ((String) (List (String)))) (SIGNATURE changeName ((Result) (Symbol) (Symbol) (Result)))) + [1] continuousAtPoint?: not known that (AlgebraicallyClosedField) + [2] continuousAtPoint?: not known that (TranscendentalFunctionCategory) + [3] functionIsOscillatory: not known that (OrderedSet) is of mode [4] singularitiesOf: str has no value --------------non extending category---------------------- .. d01AgentsPackage of cat -(CATEGORY |package| (SIGNATURE |rangeIsFinite| ((|Union| (|:| |finite| "The range is finite") (|:| |lowerInfinite| "The bottom of range is infinite") (|:| |upperInfinite| "The top of range is infinite") (|:| |bothInfinite| "Both top and bottom points are infinite") (|:| |notEvaluated| "Range not yet evaluated")) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|))))) (SIGNATURE |functionIsContinuousAtEndPoints| ((|Union| (|:| |continuous| "Continuous at the end points") (|:| |lowerSingular| "There is a singularity at the lower end point") (|:| |upperSingular| "There is a singularity at the upper end point") (|:| |bothSingular| "There are singularities at both end points") (|:| |notEvaluated| "End point continuity not yet evaluated")) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|))))) (SIGNATURE |getlo| ((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (SIGNATURE |gethi| ((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (SIGNATURE |functionIsOscillatory| ((|Float|) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|))))) (SIGNATURE |problemPoints| ((|List| (|DoubleFloat|)) (|Expression| (|DoubleFloat|)) (|Symbol|) (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (SIGNATURE |singularitiesOf| ((|Stream| (|DoubleFloat|)) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|))))) (SIGNATURE |df2st| ((|String|) (|DoubleFloat|))) (SIGNATURE |ldf2lst| ((|List| (|String|)) (|List| (|DoubleFloat|)))) (SIGNATURE |sdf2lst| ((|List| (|String|)) (|Stream| (|DoubleFloat|)))) (SIGNATURE |commaSeparate| ((|String|) (|List| (|String|)))) (SIGNATURE |changeName| ((|Result|) (|Symbol|) (|Symbol|) (|Result|)))) has no functionIsFracPolynomial? : Record(var: Symbol,fn: Expression DoubleFloat,range: Segment OrderedCompletion DoubleFloat,abserr: DoubleFloat,relerr: DoubleFloat) -> Boolean +(CATEGORY |package| (SIGNATURE |rangeIsFinite| \end{verbatim} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -35723,11 +35748,11 @@ Warning: PFO;cmult has a duplicate definition in this file finalizing nrlib FSPRMELT Warnings: - [1] F2P: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE primitiveElement ((Record (: primelt F) (: poly (List (SparseUnivariatePolynomial F))) (: prim (SparseUnivariatePolynomial F))) (List F))) (IF (has F (AlgebraicallyClosedField)) (SIGNATURE primitiveElement ((Record (: primelt F) (: pol1 (SparseUnivariatePolynomial F)) (: pol2 (SparseUnivariatePolynomial F)) (: prim (SparseUnivariatePolynomial F))) F F)) noBranch)) - [2] K2P: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE primitiveElement ((Record (: primelt F) (: poly (List (SparseUnivariatePolynomial F))) (: prim (SparseUnivariatePolynomial F))) (List F))) (IF (has F (AlgebraicallyClosedField)) (SIGNATURE primitiveElement ((Record (: primelt F) (: pol1 (SparseUnivariatePolynomial F)) (: pol2 (SparseUnivariatePolynomial F)) (: prim (SparseUnivariatePolynomial F))) F F)) noBranch)) - [3] primitiveElement: not known that (Ring) is of mode (CATEGORY package (SIGNATURE primitiveElement ((Record (: primelt F) (: poly (List (SparseUnivariatePolynomial F))) (: prim (SparseUnivariatePolynomial F))) (List F))) (IF (has F (AlgebraicallyClosedField)) (SIGNATURE primitiveElement ((Record (: primelt F) (: pol1 (SparseUnivariatePolynomial F)) (: pol2 (SparseUnivariatePolynomial F)) (: prim (SparseUnivariatePolynomial F))) F F)) noBranch)) - [4] primitiveElement: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE primitiveElement ((Record (: primelt F) (: poly (List (SparseUnivariatePolynomial F))) (: prim (SparseUnivariatePolynomial F))) (List F))) (IF (has F (AlgebraicallyClosedField)) (SIGNATURE primitiveElement ((Record (: primelt F) (: pol1 (SparseUnivariatePolynomial F)) (: pol2 (SparseUnivariatePolynomial F)) (: prim (SparseUnivariatePolynomial F))) F F)) noBranch)) - [5] F2UP: not known that (Ring) is of mode (CATEGORY $ (SIGNATURE primitiveElement ((Record (: primelt F) (: pol1 (SparseUnivariatePolynomial F)) (: pol2 (SparseUnivariatePolynomial F)) (: prim (SparseUnivariatePolynomial F))) F F))) + [1] F2P: not known that (OrderedSet) is of mode + [2] K2P: not known that (OrderedSet) is of mode + [3] primitiveElement: not known that (Ring) is of mode + [4] primitiveElement: not known that (OrderedSet) is of mode + [5] F2UP: not known that (Ring) is of mode (CATEGORY $ \end{verbatim} @@ -35770,7 +35795,7 @@ Warning: REGSET;decompose has a duplicate definition in this file --------------non extending category---------------------- .. RegularChain(#1,#2) of cat -(|Join| (|RegularTriangularSetCategory| |#1| (|IndexedExponents| (|OrderedVariableList| |#2|)) (|OrderedVariableList| |#2|) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (CATEGORY |domain| (SIGNATURE |zeroSetSplit| ((|List| $) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|Boolean|) (|Boolean|))))) has no internalAugment : (NewSparseMultivariatePolynomial(#1,OrderedVariableList #2),%,Boolean,Boolean,Boolean,Boolean,Boolean) -> List % +(|Join| (|RegularTriangularSetCategory| |#1| (|IndexedExponents| \end{verbatim} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -35817,19 +35842,58 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib RSDCMPK ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((KrullNumber (N LP Split))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((numberOfVariables (N LP Split))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((algebraicDecompose ((Record (: done Split) (: todo (List LpWT))) P TS B))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((transcendentalDecompose ((Record (: done Split) (: todo (List LpWT))) P TS N))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((transcendentalDecompose ((Record (: done Split) (: todo (List LpWT))) P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((internalDecompose ((Record (: done Split) (: todo (List LpWT))) P TS N B))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((internalDecompose ((Record (: done Split) (: todo (List LpWT))) P TS N))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((internalDecompose ((Record (: done Split) (: todo (List LpWT))) P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((decompose (Split LP Split B B))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((decompose (Split LP Split B B B B B))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((upDateBranches ((List LpWT) LP Split (List LpWT) Wip N))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((convert ((String) (Record (: val (List P)) (: tower TS))))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad-->RegularSetDecompositionPackage((printInfo ((Void) (List (Record (: val (List P)) (: tower TS))) N))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((KrullNumber (N LP Split))): +Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((numberOfVariables (N LP Split))): +Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((algebraicDecompose ((Record +(: done Split) (: todo (List LpWT))) P TS B))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((transcendentalDecompose ((Record +(: done Split) (: todo (List LpWT))) P TS N))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((transcendentalDecompose ((Record +(: done Split) (: todo (List LpWT))) P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((internalDecompose ((Record +(: done Split) (: todo (List LpWT))) P TS N B))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((internalDecompose ((Record +(: done Split) (: todo (List LpWT))) P TS N))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((internalDecompose ((Record +(: done Split) (: todo (List LpWT))) P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((decompose (Split LP Split B B))): +Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((decompose (Split LP Split B B B B B))): +Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((upDateBranches ((List LpWT) LP Split +(List LpWT) Wip N))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((convert ((String) (Record (: val +(List P)) (: tower TS))))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/RSDCMPK.spad +-->RegularSetDecompositionPackage((printInfo ((Void) (List (Record +(: val (List P)) (: tower TS))) N))): Not documented!!!! \end{verbatim} @@ -35855,8 +35919,8 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib SIGNEF Warnings: - [1] sign: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE sign ((Union (Integer) failed) F)) (SIGNATURE sign ((Union (Integer) failed) F (Symbol) (OrderedCompletion F))) (SIGNATURE sign ((Union (Integer) failed) F (Symbol) F (String)))) - [2] smpsign: not known that (IntegralDomain) is of mode (CATEGORY package (SIGNATURE sign ((Union (Integer) failed) F)) (SIGNATURE sign ((Union (Integer) failed) F (Symbol) (OrderedCompletion F))) (SIGNATURE sign ((Union (Integer) failed) F (Symbol) F (String)))) + [1] sign: not known that (OrderedSet) is of mode + [2] smpsign: not known that (IntegralDomain) is of mode \end{verbatim} @@ -35882,15 +35946,15 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib SOLVETRA Warnings: - [1] solveInner: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE solve ((List (Equation (Expression R))) (Expression R))) (SIGNATURE solve ((List (Equation (Expression R))) (Equation (Expression R)))) (SIGNATURE solve ((List (Equation (Expression R))) (Equation (Expression R)) (Symbol))) (SIGNATURE solve ((List (Equation (Expression R))) (Expression R) (Symbol))) (SIGNATURE solve ((List (List (Equation (Expression R)))) (List (Equation (Expression R))) (List (Symbol))))) - [2] solveInner: not known that (Ring) is of mode (CATEGORY package (SIGNATURE solve ((List (Equation (Expression R))) (Expression R))) (SIGNATURE solve ((List (Equation (Expression R))) (Equation (Expression R)))) (SIGNATURE solve ((List (Equation (Expression R))) (Equation (Expression R)) (Symbol))) (SIGNATURE solve ((List (Equation (Expression R))) (Expression R) (Symbol))) (SIGNATURE solve ((List (List (Equation (Expression R)))) (List (Equation (Expression R))) (List (Symbol))))) - [3] tryToTrans: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has R (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has R (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [4] tryToTrans: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has R (GcdDomain)) (PROGN (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $))) (SIGNATURE squareFreePolynomial ((Factored (SparseUnivariatePolynomial $)) (SparseUnivariatePolynomial $)))) noBranch) (IF (has R (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch)) noBranch)) - [5] subsTan: not known that (IntegralDomain) is of mode (CATEGORY package (SIGNATURE solve ((List (Equation (Expression R))) (Expression R))) (SIGNATURE solve ((List (Equation (Expression R))) (Equation (Expression R)))) (SIGNATURE solve ((List (Equation (Expression R))) (Equation (Expression R)) (Symbol))) (SIGNATURE solve ((List (Equation (Expression R))) (Expression R) (Symbol))) (SIGNATURE solve ((List (List (Equation (Expression R)))) (List (Equation (Expression R))) (List (Symbol))))) + [1] solveInner: not known that (OrderedSet) is of mode + [2] solveInner: not known that (Ring) is of mode + [3] tryToTrans: not known that (TranscendentalFunctionCategory) is of + [4] tryToTrans: not known that (AlgebraicallyClosedField) is of mode + [5] subsTan: not known that (IntegralDomain) is of mode [6] buildnexpr: anscoeff has no value [7] buildnexpr: ansmant has no value [8] combineLog: ans has no value - [9] funcinv: not known that (OrderedSet) is of mode (CATEGORY R (ATTRIBUTE complex)) + [9] funcinv: not known that (OrderedSet) is of mode \end{verbatim} @@ -35917,19 +35981,64 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib SRDCMPK Processing SquareFreeRegularSetDecompositionPackage for Browser database: ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((KrullNumber (N LP Split))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((numberOfVariables (N LP Split))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((algebraicDecompose ((Record (: done Split) (: todo (List LpWT))) P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((transcendentalDecompose ((Record (: done Split) (: todo (List LpWT))) P TS N))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((transcendentalDecompose ((Record (: done Split) (: todo (List LpWT))) P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((internalDecompose ((Record (: done Split) (: todo (List LpWT))) P TS N B))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((internalDecompose ((Record (: done Split) (: todo (List LpWT))) P TS N))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((internalDecompose ((Record (: done Split) (: todo (List LpWT))) P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((decompose (Split LP Split B B))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((decompose (Split LP Split B B B B B))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((upDateBranches ((List LpWT) LP Split (List LpWT) Wip N))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((convert ((String) (Record (: val (List P)) (: tower TS))))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad-->SquareFreeRegularSetDecompositionPackage((printInfo ((Void) (List (Record (: val (List P)) (: tower TS))) N))): Not documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((KrullNumber (N LP +-->Split))): Not documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((numberOfVariables (N LP +-->Split))): Not documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((algebraicDecompose +-->((Record (: done Split) (: todo (List LpWT))) P TS))): Not +-->documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((transcendentalDecompose +-->((Record (: done Split) (: todo (List LpWT))) P TS N))): Not +-->documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((transcendentalDecompose +-->((Record (: done Split) (: todo (List LpWT))) P TS))): Not +-->documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((internalDecompose +-->((Record (: done Split) (: todo (List LpWT))) P TS N B))): Not +-->documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((internalDecompose +-->((Record (: done Split) (: todo (List LpWT))) P TS N))): Not +-->documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((internalDecompose +-->((Record (: done Split) (: todo (List LpWT))) P TS))): Not +-->documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((decompose (Split LP Split +-->B B))): Not documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((decompose (Split LP Split +-->B B B B B))): Not documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((upDateBranches ((List +-->LpWT) LP Split (List LpWT) Wip N))): Not documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((convert ((String) (Record +-->(: val (List P)) (: tower TS))))): Not documented!!!! + +-->/research2/test0819/mnt/fedora5/../../src/algebra/SRDCMPK.spad +-->SquareFreeRegularSetDecompositionPackage((printInfo ((Void) (List +-->(Record (: val (List P)) (: tower TS))) N))): Not documented!!!! \end{verbatim} @@ -35956,7 +36065,7 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib ZDSOLVE Warnings: [1] squareFree: toSave has no value - [2] realSolve: not known that (Ring) is of mode (CATEGORY package (SIGNATURE triangSolve ((List (RegularChain R ls)) (List (Polynomial R)) (Boolean) (Boolean))) (SIGNATURE triangSolve ((List (RegularChain R ls)) (List (Polynomial R)) (Boolean))) (SIGNATURE triangSolve ((List (RegularChain R ls)) (List (Polynomial R)))) (SIGNATURE univariateSolve ((List (Record (: complexRoots (SparseUnivariatePolynomial R)) (: coordinates (List (Polynomial R))))) (RegularChain R ls))) (SIGNATURE univariateSolve ((List (Record (: complexRoots (SparseUnivariatePolynomial R)) (: coordinates (List (Polynomial R))))) (List (Polynomial R)) (Boolean) (Boolean) (Boolean))) (SIGNATURE univariateSolve ((List (Record (: complexRoots (SparseUnivariatePolynomial R)) (: coordinates (List (Polynomial R))))) (List (Polynomial R)) (Boolean) (Boolean))) (SIGNATURE univariateSolve ((List (Record (: complexRoots (SparseUnivariatePolynomial R)) (: coordinates (List (Polynomial R))))) (List (Polynomial R)) (Boolean))) (SIGNATURE univariateSolve ((List (Record (: complexRoots (SparseUnivariatePolynomial R)) (: coordinates (List (Polynomial R))))) (List (Polynomial R)))) (SIGNATURE realSolve ((List (List (RealClosure (Fraction R)))) (RegularChain R ls))) (SIGNATURE realSolve ((List (List (RealClosure (Fraction R)))) (List (Polynomial R)) (Boolean) (Boolean) (Boolean))) (SIGNATURE realSolve ((List (List (RealClosure (Fraction R)))) (List (Polynomial R)) (Boolean) (Boolean))) (SIGNATURE realSolve ((List (List (RealClosure (Fraction R)))) (List (Polynomial R)) (Boolean))) (SIGNATURE realSolve ((List (List (RealClosure (Fraction R)))) (List (Polynomial R)))) (SIGNATURE positiveSolve ((List (List (RealClosure (Fraction R)))) (RegularChain R ls))) (SIGNATURE positiveSolve ((List (List (RealClosure (Fraction R)))) (List (Polynomial R)) (Boolean) (Boolean))) (SIGNATURE positiveSolve ((List (List (RealClosure (Fraction R)))) (List (Polynomial R)) (Boolean))) (SIGNATURE positiveSolve ((List (List (RealClosure (Fraction R)))) (List (Polynomial R)))) (SIGNATURE squareFree ((List (SquareFreeRegularTriangularSet R (IndexedExponents (OrderedVariableList ls2)) (OrderedVariableList ls2) (NewSparseMultivariatePolynomial R (OrderedVariableList ls2)))) (RegularChain R ls))) (SIGNATURE convert ((NewSparseMultivariatePolynomial R (OrderedVariableList ls2)) (NewSparseMultivariatePolynomial R (OrderedVariableList ls)))) (SIGNATURE convert ((Polynomial (RealClosure (Fraction R))) (Polynomial R))) (SIGNATURE convert ((Polynomial (RealClosure (Fraction R))) (NewSparseMultivariatePolynomial R (OrderedVariableList ls2)))) (SIGNATURE convert ((SparseUnivariatePolynomial (RealClosure (Fraction R))) (SparseUnivariatePolynomial R))) (SIGNATURE convert ((List (NewSparseMultivariatePolynomial R (OrderedVariableList ls2))) (SquareFreeRegularTriangularSet R (IndexedExponents (OrderedVariableList ls2)) (OrderedVariableList ls2) (NewSparseMultivariatePolynomial R (OrderedVariableList ls2)))))) + [2] realSolve: not known that (Ring) is of mode [3] realSolve: toSave has no value [4] positiveSolve: toSave has no value [5] univariateSolve: lq2 has no value @@ -36059,23 +36168,75 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib SFRGCD Processing SquareFreeRegularTriangularSetGcdPackage for Browser database: ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((startTableGcd! ((Void) S S S))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stopTableGcd! ((Void)))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((startTableInvSet! ((Void) S S S))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stopTableInvSet! ((Void)))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stosePrepareSubResAlgo ((List LpWT) P P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stoseInternalLastSubResultant ((List PWT) P P TS B B))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stoseInternalLastSubResultant ((List PWT) (List LpWT) V B))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stoseIntegralLastSubResultant ((List PWT) P P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stoseLastSubResultant ((List PWT) P P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertible? (B P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertible?sqfreg ((List BWT) P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertibleSetsqfreg (Split P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertible?reg ((List BWT) P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertibleSetreg (Split P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertible? ((List BWT) P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertibleSet (Split P TS))): Not documented!!!! ---->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad-->SquareFreeRegularTriangularSetGcdPackage((stoseSquareFreePart ((List PWT) P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad--> +SquareFreeRegularTriangularSetGcdPackage((startTableGcd! ((Void) S S +S))): Not documented!!!! + + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stopTableGcd! + ((Void)))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((startTableInvSet! + ((Void) S S S))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stopTableInvSet! + ((Void)))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stosePrepareSubResAlgo + ((List LpWT) P P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stoseInternalLastSubResultant + ((List PWT) P P TS B B))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stoseInternalLastSubResultant + ((List PWT) (List LpWT) V B))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stoseIntegralLastSubResultant + ((List PWT) P P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stoseLastSubResultant + ((List PWT) P P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertible? + (B P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertible?sqfreg + ((List BWT) P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertibleSetsqfreg + (Split P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertible?reg + ((List BWT) P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertibleSetreg + (Split P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertible? + ((List BWT) P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stoseInvertibleSet + (Split P TS))): Not documented!!!! + +--->/research2/test0819/mnt/fedora5/../../src/algebra/SFRGCD.spad +-->SquareFreeRegularTriangularSetGcdPackage((stoseSquareFreePart + ((List PWT) P TS))): Not documented!!!! \end{verbatim} @@ -36105,12 +36266,12 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib ODEEF Warnings: - [1] solve: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE solve ((Union (Record (: particular (Vector F)) (: basis (List (Vector F)))) failed) (Matrix F) (Vector F) (Symbol))) (SIGNATURE solve ((Union (List (Vector F)) failed) (Matrix F) (Symbol))) (SIGNATURE solve ((Union (Record (: particular (Vector F)) (: basis (List (Vector F)))) failed) (List (Equation F)) (List (BasicOperator)) (Symbol))) (SIGNATURE solve ((Union (Record (: particular (Vector F)) (: basis (List (Vector F)))) failed) (List F) (List (BasicOperator)) (Symbol))) (SIGNATURE solve ((Union (Record (: particular F) (: basis (List F))) F failed) (Equation F) (BasicOperator) (Symbol))) (SIGNATURE solve ((Union (Record (: particular F) (: basis (List F))) F failed) F (BasicOperator) (Symbol))) (SIGNATURE solve ((Union F failed) (Equation F) (BasicOperator) (Equation F) (List F))) (SIGNATURE solve ((Union F failed) F (BasicOperator) (Equation F) (List F)))) + [1] solve: not known that (OrderedSet) is of mode [2] parseODE: n has no value [3] parseODE: c has no value [4] parseODE: k has no value - [5] getcoeff: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE solve ((Union (Record (: particular (Vector F)) (: basis (List (Vector F)))) failed) (Matrix F) (Vector F) (Symbol))) (SIGNATURE solve ((Union (List (Vector F)) failed) (Matrix F) (Symbol))) (SIGNATURE solve ((Union (Record (: particular (Vector F)) (: basis (List (Vector F)))) failed) (List (Equation F)) (List (BasicOperator)) (Symbol))) (SIGNATURE solve ((Union (Record (: particular (Vector F)) (: basis (List (Vector F)))) failed) (List F) (List (BasicOperator)) (Symbol))) (SIGNATURE solve ((Union (Record (: particular F) (: basis (List F))) F failed) (Equation F) (BasicOperator) (Symbol))) (SIGNATURE solve ((Union (Record (: particular F) (: basis (List F))) F failed) F (BasicOperator) (Symbol))) (SIGNATURE solve ((Union F failed) (Equation F) (BasicOperator) (Equation F) (List F))) (SIGNATURE solve ((Union F failed) F (BasicOperator) (Equation F) (List F)))) - [6] getcoeff: not known that (Ring) is of mode (CATEGORY package (SIGNATURE solve ((Union (Record (: particular (Vector F)) (: basis (List (Vector F)))) failed) (Matrix F) (Vector F) (Symbol))) (SIGNATURE solve ((Union (List (Vector F)) failed) (Matrix F) (Symbol))) (SIGNATURE solve ((Union (Record (: particular (Vector F)) (: basis (List (Vector F)))) failed) (List (Equation F)) (List (BasicOperator)) (Symbol))) (SIGNATURE solve ((Union (Record (: particular (Vector F)) (: basis (List (Vector F)))) failed) (List F) (List (BasicOperator)) (Symbol))) (SIGNATURE solve ((Union (Record (: particular F) (: basis (List F))) F failed) (Equation F) (BasicOperator) (Symbol))) (SIGNATURE solve ((Union (Record (: particular F) (: basis (List F))) F failed) F (BasicOperator) (Symbol))) (SIGNATURE solve ((Union F failed) (Equation F) (BasicOperator) (Equation F) (List F))) (SIGNATURE solve ((Union F failed) F (BasicOperator) (Equation F) (List F)))) + [5] getcoeff: not known that (OrderedSet) is of mode + [6] getcoeff: not known that (Ring) is of mode \end{verbatim} @@ -36121,7 +36282,9 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib RINTERP Processing RationalInterpolation for Browser database: ---->-->RationalInterpolation((interpolate ((Fraction (Polynomial F)) (List F) (List F) (NonNegativeInteger) (NonNegativeInteger)))): Not documented!!!! +--->-->RationalInterpolation((interpolate ((Fraction (Polynomial F)) + (List F) (List F) (NonNegativeInteger) (NonNegativeInteger)))): + Not documented!!!! \end{verbatim} @@ -36243,8 +36406,8 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib ES- Warnings: - [1] tower: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE odd? ((Boolean) S)) (SIGNATURE even? ((Boolean) S)) (SIGNATURE eval (S S (BasicOperator) (Mapping S S))) (SIGNATURE eval (S S (BasicOperator) (Mapping S (List S)))) (SIGNATURE eval (S S (List (BasicOperator)) (List (Mapping S (List S))))) (SIGNATURE eval (S S (List (BasicOperator)) (List (Mapping S S)))) (SIGNATURE eval (S S (Symbol) (Mapping S S))) (SIGNATURE eval (S S (Symbol) (Mapping S (List S)))) (SIGNATURE eval (S S (List (Symbol)) (List (Mapping S (List S))))) (SIGNATURE eval (S S (List (Symbol)) (List (Mapping S S)))) (SIGNATURE freeOf? ((Boolean) S (Symbol))) (SIGNATURE freeOf? ((Boolean) S S)) (SIGNATURE map (S (Mapping S S) (Kernel S))) (SIGNATURE kernel (S (BasicOperator) (List S))) (SIGNATURE kernel (S (BasicOperator) S)) (SIGNATURE is? ((Boolean) S (Symbol))) (SIGNATURE is? ((Boolean) S (BasicOperator))) (SIGNATURE belong? ((Boolean) (BasicOperator))) (SIGNATURE operator ((BasicOperator) (BasicOperator))) (SIGNATURE operators ((List (BasicOperator)) S)) (SIGNATURE tower ((List (Kernel S)) S)) (SIGNATURE mainKernel ((Union (Kernel S) failed) S)) (SIGNATURE height ((NonNegativeInteger) S)) (SIGNATURE distribute (S S S)) (SIGNATURE distribute (S S)) (SIGNATURE paren (S (List S))) (SIGNATURE paren (S S)) (SIGNATURE box (S (List S))) (SIGNATURE box (S S)) (SIGNATURE subst (S S (List (Kernel S)) (List S))) (SIGNATURE subst (S S (List (Equation S)))) (SIGNATURE subst (S S (Equation S))) (SIGNATURE elt (S (BasicOperator) (List S))) (SIGNATURE elt (S (BasicOperator) S S S S)) (SIGNATURE elt (S (BasicOperator) S S S)) (SIGNATURE elt (S (BasicOperator) S S)) (SIGNATURE elt (S (BasicOperator) S)) (SIGNATURE eval (S S (List S) (List S))) (SIGNATURE eval (S S S S)) (SIGNATURE eval (S S (Equation S))) (SIGNATURE eval (S S (List (Equation S)))) (SIGNATURE eval (S S (List (Kernel S)) (List S))) (SIGNATURE eval (S S (Kernel S) S)) (SIGNATURE retract ((Kernel S) S)) (SIGNATURE retractIfCan ((Union (Kernel S) failed) S))) - [2] freeOf?: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE odd? ((Boolean) S)) (SIGNATURE even? ((Boolean) S)) (SIGNATURE eval (S S (BasicOperator) (Mapping S S))) (SIGNATURE eval (S S (BasicOperator) (Mapping S (List S)))) (SIGNATURE eval (S S (List (BasicOperator)) (List (Mapping S (List S))))) (SIGNATURE eval (S S (List (BasicOperator)) (List (Mapping S S)))) (SIGNATURE eval (S S (Symbol) (Mapping S S))) (SIGNATURE eval (S S (Symbol) (Mapping S (List S)))) (SIGNATURE eval (S S (List (Symbol)) (List (Mapping S (List S))))) (SIGNATURE eval (S S (List (Symbol)) (List (Mapping S S)))) (SIGNATURE freeOf? ((Boolean) S (Symbol))) (SIGNATURE freeOf? ((Boolean) S S)) (SIGNATURE map (S (Mapping S S) (Kernel S))) (SIGNATURE kernel (S (BasicOperator) (List S))) (SIGNATURE kernel (S (BasicOperator) S)) (SIGNATURE is? ((Boolean) S (Symbol))) (SIGNATURE is? ((Boolean) S (BasicOperator))) (SIGNATURE belong? ((Boolean) (BasicOperator))) (SIGNATURE operator ((BasicOperator) (BasicOperator))) (SIGNATURE operators ((List (BasicOperator)) S)) (SIGNATURE tower ((List (Kernel S)) S)) (SIGNATURE mainKernel ((Union (Kernel S) failed) S)) (SIGNATURE height ((NonNegativeInteger) S)) (SIGNATURE distribute (S S S)) (SIGNATURE distribute (S S)) (SIGNATURE paren (S (List S))) (SIGNATURE paren (S S)) (SIGNATURE box (S (List S))) (SIGNATURE box (S S)) (SIGNATURE subst (S S (List (Kernel S)) (List S))) (SIGNATURE subst (S S (List (Equation S)))) (SIGNATURE subst (S S (Equation S))) (SIGNATURE elt (S (BasicOperator) (List S))) (SIGNATURE elt (S (BasicOperator) S S S S)) (SIGNATURE elt (S (BasicOperator) S S S)) (SIGNATURE elt (S (BasicOperator) S S)) (SIGNATURE elt (S (BasicOperator) S)) (SIGNATURE eval (S S (List S) (List S))) (SIGNATURE eval (S S S S)) (SIGNATURE eval (S S (Equation S))) (SIGNATURE eval (S S (List (Equation S)))) (SIGNATURE eval (S S (List (Kernel S)) (List S))) (SIGNATURE eval (S S (Kernel S) S)) (SIGNATURE retract ((Kernel S) S)) (SIGNATURE retractIfCan ((Union (Kernel S) failed) S))) + [1] tower: not known that (OrderedSet) is of mode + [2] freeOf?: not known that (OrderedSet) is of mode [3] eval: IN has no value [4] eval: f has no value [5] eval: s has no value @@ -36296,13 +36459,13 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib FFIELDC- Warnings: - [1] conditionP: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE order ((PositiveInteger) S)) (SIGNATURE discreteLog ((NonNegativeInteger) S)) (SIGNATURE primitive? ((Boolean) S)) (SIGNATURE createPrimitiveElement (S)) (SIGNATURE conditionP ((Union (Vector S) failed) (Matrix S))) (SIGNATURE charthRoot (S S)) (SIGNATURE differentiate (S S)) (SIGNATURE differentiate (S S (NonNegativeInteger))) (SIGNATURE init (S)) (SIGNATURE nextItem ((Union S failed) S)) (SIGNATURE discreteLog ((Union (NonNegativeInteger) failed) S S)) (SIGNATURE order ((OnePointCompletion (PositiveInteger)) S)) (SIGNATURE charthRoot ((Union S failed) S)) (SIGNATURE gcdPolynomial ((SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S)))) + [1] conditionP: not known that (Ring) is of mode [2] order: signature of lhs not unique: (PositiveInteger)S chosen [3] order: ord has no value [4] discreteLog: disc1 has no value [5] discreteLog: disclog has no value - [6] discreteLog: not known that (IntegralDomain) is of mode (CATEGORY domain (SIGNATURE order ((PositiveInteger) S)) (SIGNATURE discreteLog ((NonNegativeInteger) S)) (SIGNATURE primitive? ((Boolean) S)) (SIGNATURE createPrimitiveElement (S)) (SIGNATURE conditionP ((Union (Vector S) failed) (Matrix S))) (SIGNATURE charthRoot (S S)) (SIGNATURE differentiate (S S)) (SIGNATURE differentiate (S S (NonNegativeInteger))) (SIGNATURE init (S)) (SIGNATURE nextItem ((Union S failed) S)) (SIGNATURE discreteLog ((Union (NonNegativeInteger) failed) S S)) (SIGNATURE order ((OnePointCompletion (PositiveInteger)) S)) (SIGNATURE charthRoot ((Union S failed) S)) (SIGNATURE gcdPolynomial ((SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S)))) - [7] gcdPolynomial: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE order ((PositiveInteger) S)) (SIGNATURE discreteLog ((NonNegativeInteger) S)) (SIGNATURE primitive? ((Boolean) S)) (SIGNATURE createPrimitiveElement (S)) (SIGNATURE conditionP ((Union (Vector S) failed) (Matrix S))) (SIGNATURE charthRoot (S S)) (SIGNATURE differentiate (S S)) (SIGNATURE differentiate (S S (NonNegativeInteger))) (SIGNATURE init (S)) (SIGNATURE nextItem ((Union S failed) S)) (SIGNATURE discreteLog ((Union (NonNegativeInteger) failed) S S)) (SIGNATURE order ((OnePointCompletion (PositiveInteger)) S)) (SIGNATURE charthRoot ((Union S failed) S)) (SIGNATURE gcdPolynomial ((SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S)))) + [6] discreteLog: not known that (IntegralDomain) is of mode + [7] gcdPolynomial: not known that (Ring) is of mode \end{verbatim} @@ -36337,7 +36500,7 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib GCDDOM- Warnings: - [1] gcdPolynomial: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE gcdPolynomial ((SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S))) (SIGNATURE lcm (S (List S))) (SIGNATURE lcm (S S S)) (SIGNATURE gcd (S (List S))) (SIGNATURE gcd (S S S))) + [1] gcdPolynomial: not known that (Ring) is of mode \end{verbatim} @@ -36386,8 +36549,8 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib INS- Warnings: - [1] factor: not known that (IntegralDomain) is of mode (CATEGORY domain (SIGNATURE invmod (S S S)) (SIGNATURE powmod (S S S S)) (SIGNATURE mask (S S)) (SIGNATURE copy (S S)) (SIGNATURE rationalIfCan ((Union (Fraction (Integer)) failed) S)) (SIGNATURE rational ((Fraction (Integer)) S)) (SIGNATURE rational? ((Boolean) S)) (SIGNATURE symmetricRemainder (S S S)) (SIGNATURE bit? ((Boolean) S S)) (SIGNATURE even? ((Boolean) S)) (SIGNATURE init (S)) (SIGNATURE nextItem ((Union S failed) S)) (SIGNATURE convert ((DoubleFloat) S)) (SIGNATURE convert ((Float) S)) (SIGNATURE permutation (S S S)) (SIGNATURE factorial (S S)) (SIGNATURE binomial (S S S)) (SIGNATURE patternMatch ((PatternMatchResult (Integer) S) S (Pattern (Integer)) (PatternMatchResult (Integer) S))) (SIGNATURE convert ((Pattern (Integer)) S)) (SIGNATURE convert ((InputForm) S)) (SIGNATURE retract ((Integer) S)) (SIGNATURE retractIfCan ((Union (Integer) failed) S)) (SIGNATURE convert ((Integer) S)) (SIGNATURE differentiate (S S)) (SIGNATURE differentiate (S S (NonNegativeInteger))) (SIGNATURE positive? ((Boolean) S)) (SIGNATURE euclideanSize ((NonNegativeInteger) S)) (SIGNATURE factor ((Factored S) S)) (SIGNATURE squareFree ((Factored S) S)) (SIGNATURE prime? ((Boolean) S)) (SIGNATURE characteristic ((NonNegativeInteger)))) - [2] patternMatch: not known that (SetCategory) is of mode (CATEGORY domain (SIGNATURE invmod (S S S)) (SIGNATURE powmod (S S S S)) (SIGNATURE mask (S S)) (SIGNATURE copy (S S)) (SIGNATURE rationalIfCan ((Union (Fraction (Integer)) failed) S)) (SIGNATURE rational ((Fraction (Integer)) S)) (SIGNATURE rational? ((Boolean) S)) (SIGNATURE symmetricRemainder (S S S)) (SIGNATURE bit? ((Boolean) S S)) (SIGNATURE even? ((Boolean) S)) (SIGNATURE init (S)) (SIGNATURE nextItem ((Union S failed) S)) (SIGNATURE convert ((DoubleFloat) S)) (SIGNATURE convert ((Float) S)) (SIGNATURE permutation (S S S)) (SIGNATURE factorial (S S)) (SIGNATURE binomial (S S S)) (SIGNATURE patternMatch ((PatternMatchResult (Integer) S) S (Pattern (Integer)) (PatternMatchResult (Integer) S))) (SIGNATURE convert ((Pattern (Integer)) S)) (SIGNATURE convert ((InputForm) S)) (SIGNATURE retract ((Integer) S)) (SIGNATURE retractIfCan ((Union (Integer) failed) S)) (SIGNATURE convert ((Integer) S)) (SIGNATURE differentiate (S S)) (SIGNATURE differentiate (S S (NonNegativeInteger))) (SIGNATURE positive? ((Boolean) S)) (SIGNATURE euclideanSize ((NonNegativeInteger) S)) (SIGNATURE factor ((Factored S) S)) (SIGNATURE squareFree ((Factored S) S)) (SIGNATURE prime? ((Boolean) S)) (SIGNATURE characteristic ((NonNegativeInteger)))) + [1] factor: not known that (IntegralDomain) is of mode + [2] patternMatch: not known that (SetCategory) is of mode [3] powmod: y has no value \end{verbatim} @@ -36413,8 +36576,8 @@ Warning: REGSET;decompose has a duplicate definition in this file Warnings: [1] OMwrite: pretend(String) -- should replace by @ [2] hash: signature of lhs not unique: $$ chosen - [3] factorPolynomial: not known that (UnivariatePolynomialCategory (Integer)) is of mode (CATEGORY domain (SIGNATURE outputForm ((OutputForm) $ (OutputForm))) (SIGNATURE fmecg ($ $ (NonNegativeInteger) $ $))) - [4] gcdPolynomial: not known that (UnivariatePolynomialCategory (Integer)) is of mode (CATEGORY domain (SIGNATURE outputForm ((OutputForm) $ (OutputForm))) (SIGNATURE fmecg ($ $ (NonNegativeInteger) $ $))) + [3] factorPolynomial: not known that (UnivariatePolynomialCategory + [4] gcdPolynomial: not known that (UnivariatePolynomialCategory \end{verbatim} @@ -36498,7 +36661,7 @@ Warning: REGSET;decompose has a duplicate definition in this file --------------non extending category---------------------- .. NonNegativeInteger of cat -(|Join| (|OrderedAbelianMonoidSup|) (|Monoid|) (CATEGORY |domain| (SIGNATURE |quo| ($ $ $)) (SIGNATURE |rem| ($ $ $)) (SIGNATURE |gcd| ($ $ $)) (SIGNATURE |divide| ((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $)) (SIGNATURE |exquo| ((|Union| $ "failed") $ $)) (SIGNATURE |shift| ($ $ (|Integer|))) (SIGNATURE |random| ($ $)) (ATTRIBUTE (|commutative| "*")))) has no +(|Join| (|OrderedAbelianMonoidSup|) (|Monoid|) (CATEGORY |domain| ` \end{verbatim} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -36554,16 +36717,16 @@ Warning: REGSET;decompose has a duplicate definition in this file finalizing nrlib POLYCAT- Warnings: [1] eval: IN has no value - [2] coefficient: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE gcdPolynomial ((SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S))) (SIGNATURE squareFree ((Factored S) S)) (SIGNATURE squareFreePart (S S)) (SIGNATURE charthRoot ((Union S failed) S)) (SIGNATURE < ((Boolean) S S)) (SIGNATURE convert ((InputForm) S)) (SIGNATURE convert ((Pattern (Integer)) S)) (SIGNATURE convert ((Pattern (Float)) S)) (SIGNATURE patternMatch ((PatternMatchResult (Integer) S) S (Pattern (Integer)) (PatternMatchResult (Integer) S))) (SIGNATURE patternMatch ((PatternMatchResult (Float) S) S (Pattern (Float)) (PatternMatchResult (Float) S))) (SIGNATURE factor ((Factored S) S)) (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE factorSquareFreePolynomial ((Factored (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE solveLinearPolynomialEquation ((Union (List (SparseUnivariatePolynomial S)) failed) (List (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE conditionP ((Union (Vector S) failed) (Matrix S))) (SIGNATURE primitivePart (S S VarSet)) (SIGNATURE content (S S VarSet)) (SIGNATURE discriminant (S S VarSet)) (SIGNATURE resultant (S S S VarSet)) (SIGNATURE primitiveMonomials ((List S) S)) (SIGNATURE totalDegree ((NonNegativeInteger) S (List VarSet))) (SIGNATURE totalDegree ((NonNegativeInteger) S)) (SIGNATURE isExpt ((Union (Record (: var VarSet) (: exponent (NonNegativeInteger))) failed) S)) (SIGNATURE isTimes ((Union (List S) failed) S)) (SIGNATURE isPlus ((Union (List S) failed) S)) (SIGNATURE monomial (S S (List VarSet) (List (NonNegativeInteger)))) (SIGNATURE monomial (S S VarSet (NonNegativeInteger))) (SIGNATURE monicDivide ((Record (: quotient S) (: remainder S)) S S VarSet)) (SIGNATURE monomials ((List S) S)) (SIGNATURE coefficient (S S (List VarSet) (List (NonNegativeInteger)))) (SIGNATURE coefficient (S S VarSet (NonNegativeInteger))) (SIGNATURE reducedSystem ((Matrix R) (Matrix S))) (SIGNATURE reducedSystem ((Record (: mat (Matrix R)) (: vec (Vector R))) (Matrix S) (Vector S))) (SIGNATURE reducedSystem ((Record (: mat (Matrix (Integer))) (: vec (Vector (Integer)))) (Matrix S) (Vector S))) (SIGNATURE reducedSystem ((Matrix (Integer)) (Matrix S))) (SIGNATURE retract (VarSet S)) (SIGNATURE retractIfCan ((Union VarSet failed) S)) (SIGNATURE eval (S S (List VarSet) (List S))) (SIGNATURE eval (S S VarSet S)) (SIGNATURE eval (S S (List VarSet) (List R))) (SIGNATURE eval (S S VarSet R)) (SIGNATURE eval (S S (List S) (List S))) (SIGNATURE eval (S S S S)) (SIGNATURE eval (S S (Equation S))) (SIGNATURE eval (S S (List (Equation S)))) (SIGNATURE monomial (S R E)) (SIGNATURE coefficient (R S E)) (SIGNATURE retract ((Integer) S)) (SIGNATURE retractIfCan ((Union (Integer) failed) S)) (SIGNATURE retract ((Fraction (Integer)) S)) (SIGNATURE retractIfCan ((Union (Fraction (Integer)) failed) S)) (SIGNATURE retractIfCan ((Union R failed) S)) (SIGNATURE retract (R S)) (SIGNATURE content (R S)) (SIGNATURE primitivePart (S S))) + [2] coefficient: not known that (Ring) is of mode [3] totalDegree: w has no value [4] reducedSystem: IN has no value [5] reducedSystem: r has no value - [6] reducedSystem: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE gcdPolynomial ((SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S))) (SIGNATURE squareFree ((Factored S) S)) (SIGNATURE squareFreePart (S S)) (SIGNATURE charthRoot ((Union S failed) S)) (SIGNATURE < ((Boolean) S S)) (SIGNATURE convert ((InputForm) S)) (SIGNATURE convert ((Pattern (Integer)) S)) (SIGNATURE convert ((Pattern (Float)) S)) (SIGNATURE patternMatch ((PatternMatchResult (Integer) S) S (Pattern (Integer)) (PatternMatchResult (Integer) S))) (SIGNATURE patternMatch ((PatternMatchResult (Float) S) S (Pattern (Float)) (PatternMatchResult (Float) S))) (SIGNATURE factor ((Factored S) S)) (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE factorSquareFreePolynomial ((Factored (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE solveLinearPolynomialEquation ((Union (List (SparseUnivariatePolynomial S)) failed) (List (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE conditionP ((Union (Vector S) failed) (Matrix S))) (SIGNATURE primitivePart (S S VarSet)) (SIGNATURE content (S S VarSet)) (SIGNATURE discriminant (S S VarSet)) (SIGNATURE resultant (S S S VarSet)) (SIGNATURE primitiveMonomials ((List S) S)) (SIGNATURE totalDegree ((NonNegativeInteger) S (List VarSet))) (SIGNATURE totalDegree ((NonNegativeInteger) S)) (SIGNATURE isExpt ((Union (Record (: var VarSet) (: exponent (NonNegativeInteger))) failed) S)) (SIGNATURE isTimes ((Union (List S) failed) S)) (SIGNATURE isPlus ((Union (List S) failed) S)) (SIGNATURE monomial (S S (List VarSet) (List (NonNegativeInteger)))) (SIGNATURE monomial (S S VarSet (NonNegativeInteger))) (SIGNATURE monicDivide ((Record (: quotient S) (: remainder S)) S S VarSet)) (SIGNATURE monomials ((List S) S)) (SIGNATURE coefficient (S S (List VarSet) (List (NonNegativeInteger)))) (SIGNATURE coefficient (S S VarSet (NonNegativeInteger))) (SIGNATURE reducedSystem ((Matrix R) (Matrix S))) (SIGNATURE reducedSystem ((Record (: mat (Matrix R)) (: vec (Vector R))) (Matrix S) (Vector S))) (SIGNATURE reducedSystem ((Record (: mat (Matrix (Integer))) (: vec (Vector (Integer)))) (Matrix S) (Vector S))) (SIGNATURE reducedSystem ((Matrix (Integer)) (Matrix S))) (SIGNATURE retract (VarSet S)) (SIGNATURE retractIfCan ((Union VarSet failed) S)) (SIGNATURE eval (S S (List VarSet) (List S))) (SIGNATURE eval (S S VarSet S)) (SIGNATURE eval (S S (List VarSet) (List R))) (SIGNATURE eval (S S VarSet R)) (SIGNATURE eval (S S (List S) (List S))) (SIGNATURE eval (S S S S)) (SIGNATURE eval (S S (Equation S))) (SIGNATURE eval (S S (List (Equation S)))) (SIGNATURE monomial (S R E)) (SIGNATURE coefficient (R S E)) (SIGNATURE retract ((Integer) S)) (SIGNATURE retractIfCan ((Union (Integer) failed) S)) (SIGNATURE retract ((Fraction (Integer)) S)) (SIGNATURE retractIfCan ((Union (Fraction (Integer)) failed) S)) (SIGNATURE retractIfCan ((Union R failed) S)) (SIGNATURE retract (R S)) (SIGNATURE content (R S)) (SIGNATURE primitivePart (S S))) - [7] solveLinearPolynomialEquation: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE gcdPolynomial ((SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S))) (SIGNATURE squareFree ((Factored S) S)) (SIGNATURE squareFreePart (S S)) (SIGNATURE charthRoot ((Union S failed) S)) (SIGNATURE < ((Boolean) S S)) (SIGNATURE convert ((InputForm) S)) (SIGNATURE convert ((Pattern (Integer)) S)) (SIGNATURE convert ((Pattern (Float)) S)) (SIGNATURE patternMatch ((PatternMatchResult (Integer) S) S (Pattern (Integer)) (PatternMatchResult (Integer) S))) (SIGNATURE patternMatch ((PatternMatchResult (Float) S) S (Pattern (Float)) (PatternMatchResult (Float) S))) (SIGNATURE factor ((Factored S) S)) (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE factorSquareFreePolynomial ((Factored (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE solveLinearPolynomialEquation ((Union (List (SparseUnivariatePolynomial S)) failed) (List (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE conditionP ((Union (Vector S) failed) (Matrix S))) (SIGNATURE primitivePart (S S VarSet)) (SIGNATURE content (S S VarSet)) (SIGNATURE discriminant (S S VarSet)) (SIGNATURE resultant (S S S VarSet)) (SIGNATURE primitiveMonomials ((List S) S)) (SIGNATURE totalDegree ((NonNegativeInteger) S (List VarSet))) (SIGNATURE totalDegree ((NonNegativeInteger) S)) (SIGNATURE isExpt ((Union (Record (: var VarSet) (: exponent (NonNegativeInteger))) failed) S)) (SIGNATURE isTimes ((Union (List S) failed) S)) (SIGNATURE isPlus ((Union (List S) failed) S)) (SIGNATURE monomial (S S (List VarSet) (List (NonNegativeInteger)))) (SIGNATURE monomial (S S VarSet (NonNegativeInteger))) (SIGNATURE monicDivide ((Record (: quotient S) (: remainder S)) S S VarSet)) (SIGNATURE monomials ((List S) S)) (SIGNATURE coefficient (S S (List VarSet) (List (NonNegativeInteger)))) (SIGNATURE coefficient (S S VarSet (NonNegativeInteger))) (SIGNATURE reducedSystem ((Matrix R) (Matrix S))) (SIGNATURE reducedSystem ((Record (: mat (Matrix R)) (: vec (Vector R))) (Matrix S) (Vector S))) (SIGNATURE reducedSystem ((Record (: mat (Matrix (Integer))) (: vec (Vector (Integer)))) (Matrix S) (Vector S))) (SIGNATURE reducedSystem ((Matrix (Integer)) (Matrix S))) (SIGNATURE retract (VarSet S)) (SIGNATURE retractIfCan ((Union VarSet failed) S)) (SIGNATURE eval (S S (List VarSet) (List S))) (SIGNATURE eval (S S VarSet S)) (SIGNATURE eval (S S (List VarSet) (List R))) (SIGNATURE eval (S S VarSet R)) (SIGNATURE eval (S S (List S) (List S))) (SIGNATURE eval (S S S S)) (SIGNATURE eval (S S (Equation S))) (SIGNATURE eval (S S (List (Equation S)))) (SIGNATURE monomial (S R E)) (SIGNATURE coefficient (R S E)) (SIGNATURE retract ((Integer) S)) (SIGNATURE retractIfCan ((Union (Integer) failed) S)) (SIGNATURE retract ((Fraction (Integer)) S)) (SIGNATURE retractIfCan ((Union (Fraction (Integer)) failed) S)) (SIGNATURE retractIfCan ((Union R failed) S)) (SIGNATURE retract (R S)) (SIGNATURE content (R S)) (SIGNATURE primitivePart (S S))) - [8] factorPolynomial: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE gcdPolynomial ((SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S))) (SIGNATURE squareFree ((Factored S) S)) (SIGNATURE squareFreePart (S S)) (SIGNATURE charthRoot ((Union S failed) S)) (SIGNATURE < ((Boolean) S S)) (SIGNATURE convert ((InputForm) S)) (SIGNATURE convert ((Pattern (Integer)) S)) (SIGNATURE convert ((Pattern (Float)) S)) (SIGNATURE patternMatch ((PatternMatchResult (Integer) S) S (Pattern (Integer)) (PatternMatchResult (Integer) S))) (SIGNATURE patternMatch ((PatternMatchResult (Float) S) S (Pattern (Float)) (PatternMatchResult (Float) S))) (SIGNATURE factor ((Factored S) S)) (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE factorSquareFreePolynomial ((Factored (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE solveLinearPolynomialEquation ((Union (List (SparseUnivariatePolynomial S)) failed) (List (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE conditionP ((Union (Vector S) failed) (Matrix S))) (SIGNATURE primitivePart (S S VarSet)) (SIGNATURE content (S S VarSet)) (SIGNATURE discriminant (S S VarSet)) (SIGNATURE resultant (S S S VarSet)) (SIGNATURE primitiveMonomials ((List S) S)) (SIGNATURE totalDegree ((NonNegativeInteger) S (List VarSet))) (SIGNATURE totalDegree ((NonNegativeInteger) S)) (SIGNATURE isExpt ((Union (Record (: var VarSet) (: exponent (NonNegativeInteger))) failed) S)) (SIGNATURE isTimes ((Union (List S) failed) S)) (SIGNATURE isPlus ((Union (List S) failed) S)) (SIGNATURE monomial (S S (List VarSet) (List (NonNegativeInteger)))) (SIGNATURE monomial (S S VarSet (NonNegativeInteger))) (SIGNATURE monicDivide ((Record (: quotient S) (: remainder S)) S S VarSet)) (SIGNATURE monomials ((List S) S)) (SIGNATURE coefficient (S S (List VarSet) (List (NonNegativeInteger)))) (SIGNATURE coefficient (S S VarSet (NonNegativeInteger))) (SIGNATURE reducedSystem ((Matrix R) (Matrix S))) (SIGNATURE reducedSystem ((Record (: mat (Matrix R)) (: vec (Vector R))) (Matrix S) (Vector S))) (SIGNATURE reducedSystem ((Record (: mat (Matrix (Integer))) (: vec (Vector (Integer)))) (Matrix S) (Vector S))) (SIGNATURE reducedSystem ((Matrix (Integer)) (Matrix S))) (SIGNATURE retract (VarSet S)) (SIGNATURE retractIfCan ((Union VarSet failed) S)) (SIGNATURE eval (S S (List VarSet) (List S))) (SIGNATURE eval (S S VarSet S)) (SIGNATURE eval (S S (List VarSet) (List R))) (SIGNATURE eval (S S VarSet R)) (SIGNATURE eval (S S (List S) (List S))) (SIGNATURE eval (S S S S)) (SIGNATURE eval (S S (Equation S))) (SIGNATURE eval (S S (List (Equation S)))) (SIGNATURE monomial (S R E)) (SIGNATURE coefficient (R S E)) (SIGNATURE retract ((Integer) S)) (SIGNATURE retractIfCan ((Union (Integer) failed) S)) (SIGNATURE retract ((Fraction (Integer)) S)) (SIGNATURE retractIfCan ((Union (Fraction (Integer)) failed) S)) (SIGNATURE retractIfCan ((Union R failed) S)) (SIGNATURE retract (R S)) (SIGNATURE content (R S)) (SIGNATURE primitivePart (S S))) - [9] factor: not known that (IntegralDomain) is of mode (CATEGORY domain (SIGNATURE gcdPolynomial ((SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S))) (SIGNATURE squareFree ((Factored S) S)) (SIGNATURE squareFreePart (S S)) (SIGNATURE charthRoot ((Union S failed) S)) (SIGNATURE < ((Boolean) S S)) (SIGNATURE convert ((InputForm) S)) (SIGNATURE convert ((Pattern (Integer)) S)) (SIGNATURE convert ((Pattern (Float)) S)) (SIGNATURE patternMatch ((PatternMatchResult (Integer) S) S (Pattern (Integer)) (PatternMatchResult (Integer) S))) (SIGNATURE patternMatch ((PatternMatchResult (Float) S) S (Pattern (Float)) (PatternMatchResult (Float) S))) (SIGNATURE factor ((Factored S) S)) (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE factorSquareFreePolynomial ((Factored (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE solveLinearPolynomialEquation ((Union (List (SparseUnivariatePolynomial S)) failed) (List (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE conditionP ((Union (Vector S) failed) (Matrix S))) (SIGNATURE primitivePart (S S VarSet)) (SIGNATURE content (S S VarSet)) (SIGNATURE discriminant (S S VarSet)) (SIGNATURE resultant (S S S VarSet)) (SIGNATURE primitiveMonomials ((List S) S)) (SIGNATURE totalDegree ((NonNegativeInteger) S (List VarSet))) (SIGNATURE totalDegree ((NonNegativeInteger) S)) (SIGNATURE isExpt ((Union (Record (: var VarSet) (: exponent (NonNegativeInteger))) failed) S)) (SIGNATURE isTimes ((Union (List S) failed) S)) (SIGNATURE isPlus ((Union (List S) failed) S)) (SIGNATURE monomial (S S (List VarSet) (List (NonNegativeInteger)))) (SIGNATURE monomial (S S VarSet (NonNegativeInteger))) (SIGNATURE monicDivide ((Record (: quotient S) (: remainder S)) S S VarSet)) (SIGNATURE monomials ((List S) S)) (SIGNATURE coefficient (S S (List VarSet) (List (NonNegativeInteger)))) (SIGNATURE coefficient (S S VarSet (NonNegativeInteger))) (SIGNATURE reducedSystem ((Matrix R) (Matrix S))) (SIGNATURE reducedSystem ((Record (: mat (Matrix R)) (: vec (Vector R))) (Matrix S) (Vector S))) (SIGNATURE reducedSystem ((Record (: mat (Matrix (Integer))) (: vec (Vector (Integer)))) (Matrix S) (Vector S))) (SIGNATURE reducedSystem ((Matrix (Integer)) (Matrix S))) (SIGNATURE retract (VarSet S)) (SIGNATURE retractIfCan ((Union VarSet failed) S)) (SIGNATURE eval (S S (List VarSet) (List S))) (SIGNATURE eval (S S VarSet S)) (SIGNATURE eval (S S (List VarSet) (List R))) (SIGNATURE eval (S S VarSet R)) (SIGNATURE eval (S S (List S) (List S))) (SIGNATURE eval (S S S S)) (SIGNATURE eval (S S (Equation S))) (SIGNATURE eval (S S (List (Equation S)))) (SIGNATURE monomial (S R E)) (SIGNATURE coefficient (R S E)) (SIGNATURE retract ((Integer) S)) (SIGNATURE retractIfCan ((Union (Integer) failed) S)) (SIGNATURE retract ((Fraction (Integer)) S)) (SIGNATURE retractIfCan ((Union (Fraction (Integer)) failed) S)) (SIGNATURE retractIfCan ((Union R failed) S)) (SIGNATURE retract (R S)) (SIGNATURE content (R S)) (SIGNATURE primitivePart (S S))) + [6] reducedSystem: not known that (Ring) is of mode + [7] solveLinearPolynomialEquation: not known that (Ring) is of mode + [8] factorPolynomial: not known that (Ring) is of mode + [9] factor: not known that (IntegralDomain) is of mode [10] conditionP: :(Integer) --should replace by pretend - [11] patternMatch: not known that (SetCategory) is of mode (CATEGORY domain (SIGNATURE gcdPolynomial ((SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S) (SparseUnivariatePolynomial S))) (SIGNATURE squareFree ((Factored S) S)) (SIGNATURE squareFreePart (S S)) (SIGNATURE charthRoot ((Union S failed) S)) (SIGNATURE < ((Boolean) S S)) (SIGNATURE convert ((InputForm) S)) (SIGNATURE convert ((Pattern (Integer)) S)) (SIGNATURE convert ((Pattern (Float)) S)) (SIGNATURE patternMatch ((PatternMatchResult (Integer) S) S (Pattern (Integer)) (PatternMatchResult (Integer) S))) (SIGNATURE patternMatch ((PatternMatchResult (Float) S) S (Pattern (Float)) (PatternMatchResult (Float) S))) (SIGNATURE factor ((Factored S) S)) (SIGNATURE factorPolynomial ((Factored (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE factorSquareFreePolynomial ((Factored (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE solveLinearPolynomialEquation ((Union (List (SparseUnivariatePolynomial S)) failed) (List (SparseUnivariatePolynomial S)) (SparseUnivariatePolynomial S))) (SIGNATURE conditionP ((Union (Vector S) failed) (Matrix S))) (SIGNATURE primitivePart (S S VarSet)) (SIGNATURE content (S S VarSet)) (SIGNATURE discriminant (S S VarSet)) (SIGNATURE resultant (S S S VarSet)) (SIGNATURE primitiveMonomials ((List S) S)) (SIGNATURE totalDegree ((NonNegativeInteger) S (List VarSet))) (SIGNATURE totalDegree ((NonNegativeInteger) S)) (SIGNATURE isExpt ((Union (Record (: var VarSet) (: exponent (NonNegativeInteger))) failed) S)) (SIGNATURE isTimes ((Union (List S) failed) S)) (SIGNATURE isPlus ((Union (List S) failed) S)) (SIGNATURE monomial (S S (List VarSet) (List (NonNegativeInteger)))) (SIGNATURE monomial (S S VarSet (NonNegativeInteger))) (SIGNATURE monicDivide ((Record (: quotient S) (: remainder S)) S S VarSet)) (SIGNATURE monomials ((List S) S)) (SIGNATURE coefficient (S S (List VarSet) (List (NonNegativeInteger)))) (SIGNATURE coefficient (S S VarSet (NonNegativeInteger))) (SIGNATURE reducedSystem ((Matrix R) (Matrix S))) (SIGNATURE reducedSystem ((Record (: mat (Matrix R)) (: vec (Vector R))) (Matrix S) (Vector S))) (SIGNATURE reducedSystem ((Record (: mat (Matrix (Integer))) (: vec (Vector (Integer)))) (Matrix S) (Vector S))) (SIGNATURE reducedSystem ((Matrix (Integer)) (Matrix S))) (SIGNATURE retract (VarSet S)) (SIGNATURE retractIfCan ((Union VarSet failed) S)) (SIGNATURE eval (S S (List VarSet) (List S))) (SIGNATURE eval (S S VarSet S)) (SIGNATURE eval (S S (List VarSet) (List R))) (SIGNATURE eval (S S VarSet R)) (SIGNATURE eval (S S (List S) (List S))) (SIGNATURE eval (S S S S)) (SIGNATURE eval (S S (Equation S))) (SIGNATURE eval (S S (List (Equation S)))) (SIGNATURE monomial (S R E)) (SIGNATURE coefficient (R S E)) (SIGNATURE retract ((Integer) S)) (SIGNATURE retractIfCan ((Union (Integer) failed) S)) (SIGNATURE retract ((Fraction (Integer)) S)) (SIGNATURE retractIfCan ((Union (Fraction (Integer)) failed) S)) (SIGNATURE retractIfCan ((Union R failed) S)) (SIGNATURE retract (R S)) (SIGNATURE content (R S)) (SIGNATURE primitivePart (S S))) + [11] patternMatch: not known that (SetCategory) is of mode \end{verbatim} @@ -36634,8 +36797,8 @@ Warning: PSETCAT-;exactQuo has a duplicate definition in this file finalizing nrlib QFCAT- Warnings: - [1] reducedSystem: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE < ((Boolean) A A)) (SIGNATURE init (A)) (SIGNATURE nextItem ((Union A failed) A)) (SIGNATURE retract ((Integer) A)) (SIGNATURE retractIfCan ((Union (Integer) failed) A)) (SIGNATURE retract ((Fraction (Integer)) A)) (SIGNATURE retractIfCan ((Union (Fraction (Integer)) failed) A)) (SIGNATURE convert ((DoubleFloat) A)) (SIGNATURE convert ((Float) A)) (SIGNATURE convert ((InputForm) A)) (SIGNATURE retract ((Symbol) A)) (SIGNATURE retractIfCan ((Union (Symbol) failed) A)) (SIGNATURE coerce (A (Symbol))) (SIGNATURE random (A)) (SIGNATURE fractionPart (A A)) (SIGNATURE denominator (A A)) (SIGNATURE numerator (A A)) (SIGNATURE patternMatch ((PatternMatchResult (Float) A) A (Pattern (Float)) (PatternMatchResult (Float) A))) (SIGNATURE patternMatch ((PatternMatchResult (Integer) A) A (Pattern (Integer)) (PatternMatchResult (Integer) A))) (SIGNATURE convert ((Pattern (Float)) A)) (SIGNATURE convert ((Pattern (Integer)) A)) (SIGNATURE reducedSystem ((Matrix S) (Matrix A))) (SIGNATURE reducedSystem ((Record (: mat (Matrix S)) (: vec (Vector S))) (Matrix A) (Vector A))) (SIGNATURE reducedSystem ((Record (: mat (Matrix (Integer))) (: vec (Vector (Integer)))) (Matrix A) (Vector A))) (SIGNATURE reducedSystem ((Matrix (Integer)) (Matrix A))) (SIGNATURE differentiate (A A (Mapping S S))) (SIGNATURE differentiate (A A (Mapping S S) (NonNegativeInteger))) (SIGNATURE differentiate (A A (List (Symbol)) (List (NonNegativeInteger)))) (SIGNATURE differentiate (A A (Symbol) (NonNegativeInteger))) (SIGNATURE differentiate (A A (List (Symbol)))) (SIGNATURE differentiate (A A (Symbol))) (SIGNATURE differentiate (A A (NonNegativeInteger))) (SIGNATURE differentiate (A A)) (SIGNATURE map (A (Mapping S S) A)) (SIGNATURE retract (S A)) (SIGNATURE retractIfCan ((Union S failed) A)) (SIGNATURE coerce (A S)) (SIGNATURE coerce (A (Fraction (Integer)))) (SIGNATURE coerce (A A)) (SIGNATURE coerce (A (Integer))) (SIGNATURE characteristic ((NonNegativeInteger))) (SIGNATURE coerce ((OutputForm) A))) - [2] patternMatch: not known that (SetCategory) is of mode (CATEGORY domain (SIGNATURE < ((Boolean) A A)) (SIGNATURE init (A)) (SIGNATURE nextItem ((Union A failed) A)) (SIGNATURE retract ((Integer) A)) (SIGNATURE retractIfCan ((Union (Integer) failed) A)) (SIGNATURE retract ((Fraction (Integer)) A)) (SIGNATURE retractIfCan ((Union (Fraction (Integer)) failed) A)) (SIGNATURE convert ((DoubleFloat) A)) (SIGNATURE convert ((Float) A)) (SIGNATURE convert ((InputForm) A)) (SIGNATURE retract ((Symbol) A)) (SIGNATURE retractIfCan ((Union (Symbol) failed) A)) (SIGNATURE coerce (A (Symbol))) (SIGNATURE random (A)) (SIGNATURE fractionPart (A A)) (SIGNATURE denominator (A A)) (SIGNATURE numerator (A A)) (SIGNATURE patternMatch ((PatternMatchResult (Float) A) A (Pattern (Float)) (PatternMatchResult (Float) A))) (SIGNATURE patternMatch ((PatternMatchResult (Integer) A) A (Pattern (Integer)) (PatternMatchResult (Integer) A))) (SIGNATURE convert ((Pattern (Float)) A)) (SIGNATURE convert ((Pattern (Integer)) A)) (SIGNATURE reducedSystem ((Matrix S) (Matrix A))) (SIGNATURE reducedSystem ((Record (: mat (Matrix S)) (: vec (Vector S))) (Matrix A) (Vector A))) (SIGNATURE reducedSystem ((Record (: mat (Matrix (Integer))) (: vec (Vector (Integer)))) (Matrix A) (Vector A))) (SIGNATURE reducedSystem ((Matrix (Integer)) (Matrix A))) (SIGNATURE differentiate (A A (Mapping S S))) (SIGNATURE differentiate (A A (Mapping S S) (NonNegativeInteger))) (SIGNATURE differentiate (A A (List (Symbol)) (List (NonNegativeInteger)))) (SIGNATURE differentiate (A A (Symbol) (NonNegativeInteger))) (SIGNATURE differentiate (A A (List (Symbol)))) (SIGNATURE differentiate (A A (Symbol))) (SIGNATURE differentiate (A A (NonNegativeInteger))) (SIGNATURE differentiate (A A)) (SIGNATURE map (A (Mapping S S) A)) (SIGNATURE retract (S A)) (SIGNATURE retractIfCan ((Union S failed) A)) (SIGNATURE coerce (A S)) (SIGNATURE coerce (A (Fraction (Integer)))) (SIGNATURE coerce (A A)) (SIGNATURE coerce (A (Integer))) (SIGNATURE characteristic ((NonNegativeInteger))) (SIGNATURE coerce ((OutputForm) A))) + [1] reducedSystem: not known that (Ring) is of mode + [2] patternMatch: not known that (SetCategory) is of mode \end{verbatim} diff --git a/books/mathtools.sty b/books/mathtools.sty new file mode 100644 index 0000000..136fdb6 --- /dev/null +++ b/books/mathtools.sty @@ -0,0 +1,1650 @@ +%% +%% This is file `mathtools.sty', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mathtools.dtx (with options: `package') +%% +%% This is a generated file. +%% +%% Copyright (C) 2002-2011 by Morten Hoegholm +%% Copyright (C) 2012- by Lars Madsen +%% +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either +%% version 1.3 of this license or (at your option) any later +%% version. The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of +%% LaTeX version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status "maintained". +%% +%% This Current Maintainer of this work is +%% Lars Madsen +%% +%% This work consists of the main source file mathtools.dtx +%% and the derived files +%% mathtools.sty, mathtools.pdf, mathtools.ins, mathtools.drv. +%% +\ProvidesPackage{mathtools}% + [2015/11/12 v1.18 mathematical typesetting tools] +\RequirePackage{keyval,calc} +\RequirePackage{mhsetup}[2010/01/21] +\MHInternalSyntaxOn + % borrowed from fixltx2e +\def\EQ_MakeRobust#1{% + \@ifundefined{\expandafter\@gobble\string#1}{% + \@latex@error{The control sequence `\string#1' is undefined!% + \MessageBreak There is nothing here to make robust}% + \@eha + }% + {% + \@ifundefined{\expandafter\@gobble\string#1\space}% + {% + \expandafter\let\csname + \expandafter\@gobble\string#1\space\endcsname=#1% + \edef\reserved@a{\string#1}% + \def\reserved@b{#1}% + \edef\reserved@b{\expandafter\strip@prefix\meaning\reserved@b}% + \edef#1{% + \ifx\reserved@a\reserved@b + \noexpand\x@protect\noexpand#1% + \fi + \noexpand\protect\expandafter\noexpand + \csname\expandafter\@gobble\string#1\space\endcsname}% + }% + {\@latex@info{The control sequence `\string#1' is already robust}}% + }% +} +\def\forced_EQ_MakeRobust#1{% + \@ifundefined{\expandafter\@gobble\string#1}{% + \@latex@error{The control sequence `\string#1' is undefined!% + \MessageBreak There is nothing here to make robust}% + \@eha + }% + {% + % \@ifundefined{\expandafter\@gobble\string#1\space}% + % {% + \expandafter\let\csname + \expandafter\@gobble\string#1\space\endcsname=#1% + \edef\reserved@a{\string#1}% + \def\reserved@b{#1}% + \edef\reserved@b{\expandafter\strip@prefix\meaning\reserved@b}% + \edef#1{% + \ifx\reserved@a\reserved@b + \noexpand\x@protect\noexpand#1% + \fi + \noexpand\protect\expandafter\noexpand + \csname\expandafter\@gobble\string#1\space\endcsname}% + % }% + % {\@latex@info{The control sequence `\string#1' is already robust}}% + }% +} +\def\MT_options_name:{mathtools} +\newcommand*\mathtoolsset[1]{\setkeys{\MT_options_name:}{#1}} +\MH_new_boolean:n {fixamsmath} +\DeclareOption{fixamsmath}{ + \MH_set_boolean_T:n {fixamsmath} +} +\DeclareOption{donotfixamsmathbugs}{ + \MH_set_boolean_F:n {fixamsmath} +} +\DeclareOption{allowspaces}{ + \MH_let:NwN \MaybeMHPrecedingSpacesOff + \relax + \MH_let:NwN \MH_maybe_nospace_ifnextchar:Nnn \kernel@ifnextchar +} +\DeclareOption{disallowspaces}{ + \MH_let:NwN \MaybeMHPrecedingSpacesOff + \MHPrecedingSpacesOff + \MH_let:NwN \MH_maybe_nospace_ifnextchar:Nnn \MH_nospace_ifnextchar:Nnn +} +\MH_new_boolean:n {robustify} +\MH_set_boolean_T:n {robustify} +\DeclareOption{nonrobust}{ + \MH_set_boolean_F:n {robustify} +} +\DeclareOption*{ + \PassOptionsToPackage{\CurrentOption}{amsmath} +} +\ExecuteOptions{fixamsmath,disallowspaces} +\ProcessOptions\relax +\MHInternalSyntaxOff +\RequirePackage{amsmath}[2000/07/18] +\MHInternalSyntaxOn +\AtEndOfPackage{\MHInternalSyntaxOff} +\def\MT_true_false_error:{ + \PackageError{mathtools} + {You~ have~ to~ select~ either~ `true'~ or~ `false'} + {I'll~ assume~ you~ chose~ `false'~ for~ now.} +} +\MH_if_boolean:nT {robustify}{ + \EQ_MakeRobust\( + \EQ_MakeRobust\) + \EQ_MakeRobust\[ + \EQ_MakeRobust\] +} +\def\MT_define_tagform:nwnn #1[#2]#3#4{ + \@namedef{MT_tagform_#1:n}##1 + {\maketag@@@{#3\ignorespaces#2{##1}\unskip\@@italiccorr#4}} +} +\providecommand*\newtagform[1]{% + \@ifundefined{MT_tagform_#1:n} + {\@ifnextchar[% + {\MT_define_tagform:nwnn #1}% + {\MT_define_tagform:nwnn #1[]}% + }{\PackageError{mathtools} + {The~ tag~ form~ `#1'~ is~ already~ defined\MessageBreak + You~ probably~ want~ to~ look~ up~ \@backslashchar renewtagform~ + instead} + {I~ will~ just~ ignore~ your~ wish~ for~ now.}} +} +\newtagform{default}{(}{)} +\providecommand*\renewtagform[1]{% + \@ifundefined{MT_tagform_#1:n} + {\PackageError{mathtools} + {The~ tag~ form~ `#1'~ is~ not~ defined\MessageBreak + You~ probably~ want~ to~ look~ up~ \@backslashchar newtagform~ instead} + {I~ will~ just~ ignore~ your~ wish~ for~ now.}} + {\@ifnextchar[% + {\MT_define_tagform:nwnn #1}% + {\MT_define_tagform:nwnn #1[]}% + } +} +\providecommand*\usetagform[1]{% + \@ifundefined{MT_tagform_#1:n} + { + \PackageError{mathtools}{% + You~ have~ chosen~ the~ tag~ form~ `#1'\MessageBreak + but~ it~ appears~ to~ be~ undefined} + {I~ will~ use~ the~ default~ tag~ form~ instead.}% + \@namedef{tagform@}{\@nameuse{MT_tagform_default:n}} + } + { \@namedef{tagform@}{\@nameuse{MT_tagform_#1:n}} } + \MH_if_boolean:nT {show_only_refs}{ + \MH_let:NwN \MT_prev_tagform:n \tagform@ + \def\tagform@##1{\MT_extended_tagform:n {##1}} + } +} +\MH_new_boolean:n {manual_tag} +\MH_new_boolean:n {raw_maketag} +\MH_let:NwN \MT_AmS_tag_in_align: \tag@in@align +\def\tag@in@align{ + \global\MH_set_boolean_T:n {manual_tag} + \MT_AmS_tag_in_align: +} +\def\tag@in@display#1#{ + \relax + \global\MH_set_boolean_T:n {manual_tag} + \tag@in@display@a{#1} +} +\def\MT_extended_tagform:n #1{ + \MH_set_boolean_F:n {raw_maketag} + \if_meaning:NN \df@label\@empty + \MH_if_boolean:nTF {manual_tag}% this was \MH_if_boolean:nT before + { \MH_if_boolean:nTF {show_manual_tags} + { \MT_prev_tagform:n {#1} } + { \stepcounter{equation} } + }{\kern1sp}% this last {\kern1sp} is new. + \else: + \MH_if_boolean:nTF {manual_tag} + { \MH_if_boolean:nTF {show_manual_tags} + { \MT_prev_tagform:n {#1} } + { \@safe@activestrue + \@ifundefined{MT_r_\df@label} + { \global\MH_set_boolean_F:n {manual_tag} } + { \MT_prev_tagform:n {#1} } + \@safe@activesfalse + } + } + { + \@safe@activestrue + \@ifundefined{MT_r_\df@label} + { } + { \refstepcounter{equation}\MT_prev_tagform:n {#1} } + \@safe@activesfalse + } + \fi: + \global\MH_set_boolean_T:n {raw_maketag} +} +\def\MT_extended_maketag:n #1{ + \ifx\df@label\@empty + \MT_maketag:n {#1} + \else: + \MH_if_boolean:nTF {raw_maketag} + { + \MH_if_boolean:nTF {show_manual_tags} + { \MT_maketag:n {#1} } + { \@safe@activestrue + \@ifundefined{MT_r_\df@label} + { } + { \MT_maketag:n {#1} } + \@safe@activesfalse + } + } + { \MT_maketag:n {#1} } + \fi: + \global\MH_set_boolean_F:n {manual_tag} +} +\def\MT_extended_eqref:n #1{ + \protected@write\@auxout{} + {\string\MT@newlabel{#1}} + \textup{\let\df@label\@empty\MT_prev_tagform:n {\ref{#1}}} +} +\EQ_MakeRobust\MT_extended_eqref:n +\newcommand*\refeq[1]{ + \textup{\ref{#1}} +} +\def\MT_extended_refeq:n #1{ + \protected@write\@auxout{} + {\string\MT@newlabel{#1}} + \textup{\ref{#1}} +} +\newcommand*\MT@newlabel[1]{ \global\@namedef{MT_r_#1}{} } +\MH_new_boolean:n {show_only_refs} +\MH_new_boolean:n {show_manual_tags} +\define@key{\MT_options_name:}{showmanualtags}[true]{ + \@ifundefined{boolean_show_manual_tags_#1:} + { \MT_true_false_error: + \@nameuse{boolean_show_manual_tags_false:} + } + { \@nameuse{boolean_show_manual_tags_#1:} } +} +\newcommand*\MT_showonlyrefs_true:{ + \MH_if_boolean:nF {show_only_refs}{ + \MH_set_boolean_T:n {show_only_refs} + \MH_let:NwN \MT_incr_eqnum: \incr@eqnum + \MH_let:NwN \incr@eqnum \@empty + \MH_let:NwN \MT_array_parbox_restore: \@arrayparboxrestore + \@xp\def\@xp\@arrayparboxrestore\@xp{\@arrayparboxrestore + \MH_let:NwN \incr@eqnum \@empty + } + \MH_let:NwN \MT_prev_tagform:n \tagform@ + \MH_let:NwN \MT_eqref:n \eqref + \MH_let:NwN \MT_refeq:n \refeq + \MH_let:NwN \MT_maketag:n \maketag@@@ + \MH_let:NwN \maketag@@@ \MT_extended_maketag:n + \def\tagform@##1{\MT_extended_tagform:n {##1}} + \MH_let:NwN \eqref \MT_extended_eqref:n + \MH_let:NwN \refeq \MT_extended_refeq:n + } +} +\def\MT_showonlyrefs_false: { + \MH_if_boolean:nT {show_only_refs}{ + \MH_set_boolean_F:n {show_only_refs} + \MH_let:NwN \tagform@ \MT_prev_tagform:n + \MH_let:NwN \eqref \MT_eqref:n + \MH_let:NwN \refeq \MT_refeq:n + \MH_let:NwN \maketag@@@ \MT_maketag:n + \MH_let:NwN \incr@eqnum \MT_incr_eqnum: + \MH_let:NwN \@arrayparboxrestore \MT_array_parbox_restore: + } +} +\define@key{\MT_options_name:}{showonlyrefs}[true]{ + \@nameuse{MT_showonlyrefs_#1:} +} +\renewcommand\nonumber{ + \if@eqnsw + \if_meaning:NN \incr@eqnum\@empty + \MH_if_boolean:nF {show_only_refs} + {\addtocounter{equation}\m@ne} + \fi: + \fi: + \MH_let:NwN \print@eqnum\@empty \MH_let:NwN \incr@eqnum\@empty + \global\@eqnswfalse +} +\MHInternalSyntaxOff +\newcommand\noeqref[1]{\@bsphack + \@for\@tempa:=#1\do{% + \@safe@activestrue% + \edef\@tempa{\expandafter\@firstofone\@tempa}% + \@ifundefined{r@\@tempa}{% + \protect\G@refundefinedtrue% + \@latex@warning{Reference `\@tempa' on page \thepage \space + undefined (\string\noeqref)}% + }{}% + \if@filesw\protected@write\@auxout{}% + {\string\MT@newlabel{\@tempa}}\fi% + \@safe@activesfalse} + \@esphack} + +\providecommand\@safe@activestrue{}% +\providecommand\@safe@activesfalse{}% + +\MHInternalSyntaxOn +\providecommand*\xleftrightarrow[2][]{% + \ext@arrow 3095\MT_leftrightarrow_fill:{#1}{#2}} +\def\MT_leftrightarrow_fill:{% + \arrowfill@\leftarrow\relbar\rightarrow} +\providecommand*\xLeftarrow[2][]{% + \ext@arrow 0055{\Leftarrowfill@}{#1}{#2}} +\providecommand*\xRightarrow[2][]{% + \ext@arrow 0055{\Rightarrowfill@}{#1}{#2}} +\providecommand*\xLeftrightarrow[2][]{% + \ext@arrow 0055{\Leftrightarrowfill@}{#1}{#2}} +\def\MT_rightharpoondown_fill:{% + \arrowfill@\relbar\relbar\rightharpoondown} +\def\MT_rightharpoonup_fill:{% + \arrowfill@\relbar\relbar\rightharpoonup} +\def\MT_leftharpoondown_fill:{% + \arrowfill@\leftharpoondown\relbar\relbar} +\def\MT_leftharpoonup_fill:{% + \arrowfill@\leftharpoonup\relbar\relbar} +\providecommand*\xrightharpoondown[2][]{% + \ext@arrow 0359\MT_rightharpoondown_fill:{#1}{#2}} +\providecommand*\xrightharpoonup[2][]{% + \ext@arrow 0359\MT_rightharpoonup_fill:{#1}{#2}} +\providecommand*\xleftharpoondown[2][]{% + \ext@arrow 3095\MT_leftharpoondown_fill:{#1}{#2}} +\providecommand*\xleftharpoonup[2][]{% + \ext@arrow 3095\MT_leftharpoonup_fill:{#1}{#2}} +\providecommand*\xleftrightharpoons[2][]{\mathrel{% + \raise.22ex\hbox{% + $\ext@arrow 3095\MT_leftharpoonup_fill:{\phantom{#1}}{#2}$}% + \setbox0=\hbox{% + $\ext@arrow 0359\MT_rightharpoondown_fill:{#1}{\phantom{#2}}$}% + \kern-\wd0 \lower.22ex\box0}} +\providecommand*\xrightleftharpoons[2][]{\mathrel{% + \raise.22ex\hbox{% + $\ext@arrow 0359\MT_rightharpoonup_fill:{\phantom{#1}}{#2}$}% + \setbox0=\hbox{% + $\ext@arrow 3095\MT_leftharpoondown_fill:{#1}{\phantom{#2}}$}% + \kern-\wd0 \lower.22ex\box0}} +\providecommand*\xhookleftarrow[2][]{% + \ext@arrow 3095\MT_hookleft_fill:{#1}{#2}} +\def\MT_hookleft_fill:{% + \arrowfill@\leftarrow\relbar{\relbar\joinrel\rhook}} +\providecommand*\xhookrightarrow[2][]{% + \ext@arrow 3095\MT_hookright_fill:{#1}{#2}} +\def\MT_hookright_fill:{% + \arrowfill@{\lhook\joinrel\relbar}\relbar\rightarrow} +\providecommand*\xmapsto[2][]{% + \ext@arrow 0395\MT_mapsto_fill:{#1}{#2}} +\def\MT_mapsto_fill:{% + \arrowfill@{\mapstochar\relbar}\relbar\rightarrow} +\providecommand*\underbracket{ + \@ifnextchar[ + {\MT_underbracket_I:w} + {\MT_underbracket_I:w[\l_MT_bracketheight_fdim]}} +\def\MT_underbracket_I:w[#1]{ + \@ifnextchar[ + {\MT_underbracket_II:w[#1]} + {\MT_underbracket_II:w[#1][.7\fontdimen5\textfont2]}} +\def\MT_underbracket_II:w[#1][#2]#3{% + \mathop{\vtop{\m@th\ialign{## + \crcr + $\hfil\displaystyle{#3}\hfil$% + \crcr + \noalign{\kern .2\fontdimen5\textfont2 \nointerlineskip}% + \upbracketfill {#1}{#2}% + \crcr}}} + \limits} +\def\upbracketfill#1#2{% + \sbox\z@{$\braceld$} + \edef\l_MT_bracketheight_fdim{\the\ht\z@}% + \upbracketend{#1}{#2} + \leaders \vrule \@height \z@ \@depth #1 \hfill + \upbracketend{#1}{#2}% +} +\def\upbracketend#1#2{\vrule \@height #2 \@width #1\relax} +\providecommand*\overbracket{ + \@ifnextchar[ + {\MT_overbracket_I:w} + {\MT_overbracket_I:w[\l_MT_bracketheight_fdim]}} +\def\MT_overbracket_I:w[#1]{ + \@ifnextchar[ + {\MT_overbracket_II:w[#1]} + {\MT_overbracket_II:w[#1][.7\fontdimen5\textfont2]}} +\def\MT_overbracket_II:w[#1][#2]#3{% + \mathop{\vbox{\m@th\ialign{## + \crcr + \downbracketfill{#1}{#2}% + \crcr + \noalign{\kern .2\fontdimen5\textfont2 \nointerlineskip}% + $\hfil\displaystyle{#3}\hfil$ + \crcr}}}% + \limits} +\def\downbracketfill#1#2{% + \sbox\z@{$\braceld$}\edef\l_MT_bracketheight_fdim{\the\ht\z@} + \downbracketend{#1}{#2} + \leaders \vrule \@height #1 \@depth \z@ \hfill + \downbracketend{#1}{#2}% +} +\def\downbracketend#1#2{\vrule \@width #1\@depth #2\relax} +\MH_let:NwN \LaTeXunderbrace \underbrace +\def\underbrace#1{\mathop{\vtop{\m@th\ialign{##\crcr + $\hfil\displaystyle{#1}\hfil$\crcr + \noalign{\kern.7\fontdimen5\textfont2\nointerlineskip}% + \upbracefill\crcr\noalign{\kern.5\fontdimen5\textfont2}}}}\limits} +\MH_let:NwN \LaTeXoverbrace \overbrace +\def\overbrace#1{\mathop{\vbox{\m@th\ialign{##\crcr + \noalign{\kern.5\fontdimen5\textfont2}% + \downbracefill\crcr + \noalign{\kern.7\fontdimen5\textfont2\nointerlineskip}% + $\hfil\displaystyle{#1}\hfil$\crcr}}}\limits} +\providecommand*\lparen{(} +\providecommand*\rparen{)} + +\def\vcentcolon{\mathrel{\mathop\ordinarycolon}} +\providecommand\ordinarycolon{:} +\begingroup + \catcode`\:=\active + \lowercase{\endgroup +\def\MT_activate_colon{% + \ifnum\mathcode`\:=32768\relax + \let\ordinarycolon= :% + \else + \mathchardef\ordinarycolon\mathcode`\: % + \fi + \let :\vcentcolon + } +} +\MH_new_boolean:n {center_colon} +\define@key{\MT_options_name:}{centercolon}[true]{ + \@ifundefined{MT_active_colon_#1:} + { \MT_true_false_error:n + \@nameuse{MT_active_colon_false:} + } + { \@nameuse{MT_active_colon_#1:} } +} +\def\MT_active_colon_true: { + \MT_activate_colon + \MH_if_boolean:nF {center_colon}{ + \MH_set_boolean_T:n {center_colon} + \edef\MT_active_colon_false: + {\mathcode`\noexpand\:=\the\mathcode`\:\relax} + \mathcode`\:=32768 + } +} +\AtBeginDocument{ + \providecommand*\dblcolon{\vcentcolon\mathrel{\mkern-.9mu}\vcentcolon} + \providecommand*\coloneqq{\vcentcolon\mathrel{\mkern-1.2mu}=} + \providecommand*\Coloneqq{\dblcolon\mathrel{\mkern-1.2mu}=} + \providecommand*\coloneq{\vcentcolon\mathrel{\mkern-1.2mu}\mathrel{-}} + \providecommand*\Coloneq{\dblcolon\mathrel{\mkern-1.2mu}\mathrel{-}} + \providecommand*\eqqcolon{=\mathrel{\mkern-1.2mu}\vcentcolon} + \providecommand*\Eqqcolon{=\mathrel{\mkern-1.2mu}\dblcolon} + \providecommand*\eqcolon{\mathrel{-}\mathrel{\mkern-1.2mu}\vcentcolon} + \providecommand*\Eqcolon{\mathrel{-}\mathrel{\mkern-1.2mu}\dblcolon} + \providecommand*\colonapprox{\vcentcolon\mathrel{\mkern-1.2mu}\approx} + \providecommand*\Colonapprox{\dblcolon\mathrel{\mkern-1.2mu}\approx} + \providecommand*\colonsim{\vcentcolon\mathrel{\mkern-1.2mu}\sim} + \providecommand*\Colonsim{\dblcolon\mathrel{\mkern-1.2mu}\sim} +} +\let \AMS@math@cr@@ \math@cr@@ +\MH_new_boolean:n {mult_firstline} +\MH_new_boolean:n {outer_mult} +\newcount\g_MT_multlinerow_int +\newdimen\l_MT_multwidth_dim +\newcommand*\MT_test_for_tcb_other:nnnnn [1]{ + \if:w t#1\relax + \expandafter\MH_use_choice_i:nnnn + \else: + \if:w c#1\relax + \expandafter\expandafter\expandafter\MH_use_choice_ii:nnnn + \else: + \if:w b#1\relax + \expandafter\expandafter\expandafter + \expandafter\expandafter\expandafter\expandafter + \MH_use_choice_iii:nnnn + \else: + \expandafter\expandafter\expandafter + \expandafter\expandafter\expandafter\expandafter + \MH_use_choice_iv:nnnn + \fi: + \fi: + \fi: +} +\def\MT_mult_invisible_line: { + \crcr + \global\MH_set_boolean_F:n {mult_firstline} + \hbox to \l_MT_multwidth_dim{}\crcr + \noalign{\vskip-\baselineskip \vskip-\jot \vskip-\normallineskip} +} +\def\MT_mult_mathcr_atat:w [#1]{% + \if_num:w 0=`{\fi: \iffalse}\fi: + \MH_if_boolean:nT {mult_firstline}{ + \kern\l_MT_mult_left_fdim + \MT_mult_invisible_line: + } + \crcr + \noalign{\vskip#1\relax} + \global\advance\g_MT_multlinerow_int\@ne + \if_num:w \g_MT_multlinerow_int=\l_MT_multline_lastline_fint + \MH_let:NwN \math@cr@@\MT_mult_last_mathcr:w + \fi: +} +\def\MT_mult_firstandlast_mathcr:w [#1]{% + \if_num:w 0=`{\fi: \iffalse}\fi: + \kern\l_MT_mult_left_fdim + \MT_mult_invisible_line: + \noalign{\vskip#1\relax} + \kern\l_MT_mult_right_fdim +} +\def\MT_mult_last_mathcr:w [#1]{ + \if_num:w 0=`{\fi: \iffalse}\fi:\math@cr@@@ + \noalign{\vskip#1\relax} + \kern\l_MT_mult_right_fdim} +\newcommand\MT_start_mult:N [1]{ + \MT_test_for_tcb_other:nnnnn {#1} + { \MH_let:NwN \MT_next:\vtop } + { \MH_let:NwN \MT_next:\vcenter } + { \MH_let:NwN \MT_next:\vbox } + { + \PackageError{mathtools} + {Invalid~ position~ specifier.~ I'll~ try~ to~ recover~ with~ + `c'}\@ehc + } + \collect@body\MT_mult_internal:n +} +\newcommand*\MT_shoveright:wn [2][0pt]{% + #2\hfilneg + \setlength\@tempdima{#1} + \kern\@tempdima +} +\newcommand*\MT_shoveleft:wn [2][0pt]{% + \hfilneg + \setlength\@tempdima{#1} + \kern\@tempdima + #2 +} +\newcommand*\MT_mult_internal:n [1]{ + \MH_if_boolean:nF {outer_mult}{\null\,} + \MT_next: + \bgroup + \Let@ + \def\l_MT_multline_lastline_fint{0 } + \chardef\dspbrk@context\@ne \restore@math@cr + \MH_let:NwN \math@cr@@\MT_mult_mathcr_atat:w + \MH_let:NwN \shoveleft\MT_shoveleft:wn + \MH_let:NwN \shoveright\MT_shoveright:wn + \spread@equation + \MH_set_boolean_F:n {mult_firstline} + \MT_measure_mult:n {#1} + \if_dim:w \l_MT_multwidth_dim<\l_MT_multline_measure_fdim + \MH_setlength:dn \l_MT_multwidth_dim{\l_MT_multline_measure_fdim} + \fi + \MH_set_boolean_T:n {mult_firstline} + \if_num:w \l_MT_multline_lastline_fint=\@ne + \MH_let:NwN \math@cr@@ \MT_mult_firstandlast_mathcr:w + \fi: + \ialign\bgroup + \hfil\strut@$\m@th\displaystyle{}##$\hfil + \crcr + \hfilneg + #1 +} +\newcommand\MT_measure_mult:n [1]{ + \begingroup + \measuring@true + \g_MT_multlinerow_int\@ne + \MH_let:NwN \label\MT_gobblelabel:w + \MH_let:NwN \tag\gobble@tag + \setbox\z@\vbox{ + \ialign{\strut@$\m@th\displaystyle{}##$ + \crcr + #1 + \crcr + } + } + \xdef\l_MT_multline_measure_fdim{\the\wdz@} + \advance\g_MT_multlinerow_int\m@ne + \xdef\l_MT_multline_lastline_fint{\number\g_MT_multlinerow_int} + \endgroup + \g_MT_multlinerow_int\@ne +} +\MaybeMHPrecedingSpacesOff +\newcommand*\MT_multlined_second_arg:w [1][\@empty]{ + \MT_test_for_tcb_other:nnnnn {#1} + {\def\MT_mult_default_pos:{#1}} + {\def\MT_mult_default_pos:{#1}} + {\def\MT_mult_default_pos:{#1}} + { + \if_meaning:NN \@empty#1\@empty + \else: + \setlength \l_MT_multwidth_dim{#1} + \fi: + } + \MT_start_mult:N \MT_mult_default_pos: +} +\newcommand\MultlinedHook{ + \renewenvironment{subarray}[1]{% + \vcenter\bgroup + \Let@ \restore@math@cr \default@tag + \let\math@cr@@\AMS@math@cr@@ + \baselineskip\fontdimen10 \scriptfont\tw@ + \advance\baselineskip\fontdimen12 \scriptfont\tw@ + \lineskip\thr@@\fontdimen8 \scriptfont\thr@@ + \lineskiplimit\lineskip + \ialign\bgroup\ifx c##1\hfil\fi + $\m@th\scriptstyle####$\hfil\crcr + }{% + \crcr\egroup\egroup + } +} + +\newenvironment{multlined}[1][] + {\MH_group_align_safe_begin: + \MultlinedHook + \MT_test_for_tcb_other:nnnnn {#1} + {\def\MT_mult_default_pos:{#1}} + {\def\MT_mult_default_pos:{#1}} + {\def\MT_mult_default_pos:{#1}} + { + \if_meaning:NN \@empty#1\@empty + \else: + \setlength \l_MT_multwidth_dim{#1} + \fi: + } + \MT_multlined_second_arg:w + } + { + \hfilneg \endaligned \MH_group_align_safe_end: + } +\MHPrecedingSpacesOn +\define@key{\MT_options_name:} + {firstline-afterskip}{\def\l_MT_mult_left_fdim{#1}} +\define@key{\MT_options_name:} + {lastline-preskip}{\def\l_MT_mult_right_fdim{#1}} +\define@key{\MT_options_name:} + {multlined-width}{\setlength \l_MT_multwidth_dim{#1}} +\define@key{\MT_options_name:} + {multlined-pos}{\def\MT_mult_default_pos:{#1}} +\setkeys{\MT_options_name:}{ + firstline-afterskip=\multlinegap, + lastline-preskip=\multlinegap, + multlined-width=0pt, + multlined-pos=c, +} +\def\MT_gobblelabel:w #1{} +\newcommand\MT_delim_default_inner_wrappers:n [1]{ + \@namedef{MT_delim_\MH_cs_to_str:N #1 _star_wrapper:nnn}##1##2##3{ + \mathopen{}\mathclose\bgroup ##1 ##2 \aftergroup\egroup ##3 + } + \@namedef{MT_delim_\MH_cs_to_str:N #1 _nostar_wrapper:nnn}##1##2##3{ + \mathopen{##1}##2\mathclose{##3} + } + } + +\newcommand\reDeclarePairedDelimiterInnerWrapper[3]{ + \@namedef{MT_delim_\MH_cs_to_str:N #1 _ #2 _wrapper:nnn}##1##2##3{ + #3 + } +} + +\newcommand*\DeclarePairedDelimiter[3]{% + \@ifdefinable{#1}{ + \MT_delim_default_inner_wrappers:n{#1} + \@namedef{MT_delim_\MH_cs_to_str:N #1 _star:}##1 + %{\mathopen{}\mathclose\bgroup\left#2 ##1 \aftergroup\egroup\right #3}% + { \@nameuse{MT_delim_\MH_cs_to_str:N #1 _star_wrapper:nnn}% + {\left#2}{##1}{\right#3} }% + \@xp\@xp\@xp + \newcommand + \@xp\csname MT_delim_\MH_cs_to_str:N #1 _nostar:\endcsname + [2][\\@gobble] + { + %\mathopen{\@nameuse {\MH_cs_to_str:N ##1 l} #2} ##2 + %\mathclose{\@nameuse {\MH_cs_to_str:N ##1 r} #3}} + \@nameuse{MT_delim_\MH_cs_to_str:N #1 _nostar_wrapper:nnn}% + {\@nameuse {\MH_cs_to_str:N ##1 l} #2} + {##2} + {\@nameuse {\MH_cs_to_str:N ##1 r} #3} + } + \DeclareRobustCommand{#1}{ + \@ifstar + {\@nameuse{MT_delim_\MH_cs_to_str:N #1 _star:}} + {\@nameuse{MT_delim_\MH_cs_to_str:N #1 _nostar:}} + } + } +} +\def\MHempty{} +\def\DeclarePairedDelimiterX#1[#2]#3#4#5{% + \@ifdefinable{#1}{ + \MT_paired_delimx_arg_test:n{#2} + \MT_delim_default_inner_wrappers:n{#1} + \@xp\@xp\@xp + \newcommand + \@xp\csname MT_delim_\MH_cs_to_str:N #1 _star:\endcsname + [#2] + { + \begingroup + \def\delimsize{\middle} + %\mathopen{}\mathclose\bgroup\left#3 #5 \aftergroup\egroup\right#4 + \@nameuse{MT_delim_\MH_cs_to_str:N #1 _star_wrapper:nnn} + {\left#3}{#5}{\right#4} + \endgroup + } + \@xp\@xp\@xp + \newcommand + \@xp\csname MT_delim_\MH_cs_to_str:N #1 _nostar:\endcsname + [1][\MHempty] + { + \begingroup + \def\delimsize{##1} + \@nameuse{MT_delim_\MH_cs_to_str:N #1 _nostar_inner:} + } + \@xp\@xp\@xp + \newcommand + \@xp\csname MT_delim_\MH_cs_to_str:N #1 _nostar_inner:\endcsname + [#2] + { + %\mathopen{% + % \let\MHempty\@gobble + % \@xp\@xp\@xp\csname\@xp\MH_cs_to_str:N \delimsize l\endcsname #3} + %#5 + %\mathclose{% + % \let\MHempty\@gobble + % \@xp\@xp\@xp\csname\@xp\MH_cs_to_str:N \delimsize r\endcsname #4} + \@nameuse{MT_delim_\MH_cs_to_str:N #1 _nostar_wrapper:nnn} + { + \let\MHempty\@gobble + \@xp\@xp\@xp\csname\@xp\MH_cs_to_str:N \delimsize l\endcsname #3 + } + {#5} + { + \let\MHempty\@gobble + \@xp\@xp\@xp\csname\@xp\MH_cs_to_str:N \delimsize r\endcsname #4 + } + \endgroup + } + \DeclareRobustCommand{#1}{ + \@ifstar + {\@nameuse{MT_delim_\MH_cs_to_str:N #1 _star:}} + {\@nameuse{MT_delim_\MH_cs_to_str:N #1 _nostar:}} + } + } +} +\def\MT_paired_delimx_arg_test:n #1{ + \ifnum#1>9\relax + \PackageError{mathtools}{No~ more~ than~ 9~ arguments}{} + \else + \ifnum#1<1\relax + \PackageError{mathtools}{Macro~ need~ 1~ or~ more~ + arguments.\MessageBreak Please~ change~ [#1]~ to~ [1]~ ... [9]}{} + \fi + \fi + } + +\def\DeclarePairedDelimiterXPP#1[#2]#3#4#5#6#7{% + \@ifdefinable{#1}{ + \MT_paired_delimx_arg_test:n{#2} + \MT_delim_default_inner_wrappers:n{#1} + \@xp\@xp\@xp + \newcommand + \@xp\csname MT_delim_\MH_cs_to_str:N #1 _star:\endcsname + [#2] + { + \begingroup + \def\delimsize{\middle} + #3 + \@nameuse{MT_delim_\MH_cs_to_str:N #1 _star_wrapper:nnn} + {\left#4}{#7}{\right#5} + #6 + \endgroup + } + \@xp\@xp\@xp + \newcommand + \@xp\csname MT_delim_\MH_cs_to_str:N #1 _nostar:\endcsname + [1][\MHempty] + { + \begingroup + \def\delimsize{##1} + \@nameuse{MT_delim_\MH_cs_to_str:N #1 _nostar_inner:} + } + \@xp\@xp\@xp + \newcommand + \@xp\csname MT_delim_\MH_cs_to_str:N #1 _nostar_inner:\endcsname + [#2] + { + #3 + \@nameuse{MT_delim_\MH_cs_to_str:N #1 _nostar_wrapper:nnn} + { + \let\MHempty\@gobble + \@xp\@xp\@xp\csname\@xp\MH_cs_to_str:N \delimsize l\endcsname #4 + } + {#7} + { + \let\MHempty\@gobble + \@xp\@xp\@xp\csname\@xp\MH_cs_to_str:N \delimsize r\endcsname #5 + } + #6 + \endgroup + } + \DeclareRobustCommand{#1}{ + \@ifstar + {\@nameuse{MT_delim_\MH_cs_to_str:N #1 _star:}} + {\@nameuse{MT_delim_\MH_cs_to_str:N #1 _nostar:}} + } + } +} + + +\def\MT_start_cases:nnnn #1#2#3#4{ % #1=sep,#2=lpreamble,#3=rpreamble,#4=delim + \RIfM@\else + \nonmatherr@{\begin{\@currenvir}} + \fi + \MH_group_align_safe_begin: + \left#4 + \vcenter \bgroup + \Let@ \chardef\dspbrk@context\@ne \restore@math@cr + \let \math@cr@@\AMS@math@cr@@ + \spread@equation + \ialign\bgroup + \strut@#2 \strut@ + #3 + \crcr +} +\def\MH_end_cases:{\crcr\egroup + \restorecolumn@ + \egroup + \MH_group_align_safe_end: +} +\newcommand*\newcases[6]{% #1=name, #2=sep, #3=preamble, #4=left, #5=right + \newenvironment{#1} + {\MT_start_cases:nnnn {#2}{#3}{#4}{#5}} + {\MH_end_cases:\right#6} +} +\newcommand*\renewcases[6]{ + \renewenvironment{#1} + {\MT_start_cases:nnnn {#2}{#3}{#4}{#5}} + {\MH_end_cases:\right#6} +} +\newcases{dcases}{\quad}{% + $\m@th\displaystyle{##}$\hfil}{$\m@th\displaystyle{##}$\hfil}{\lbrace}{.} +\newcases{dcases*}{\quad}{% + $\m@th\displaystyle{##}$\hfil}{{##}\hfil}{\lbrace}{.} +\newcases{rcases}{\quad}{% + $\m@th{##}$\hfil}{$\m@th{##}$\hfil}{.}{\rbrace} +\newcases{rcases*}{\quad}{% + $\m@th{##}$\hfil}{{##}\hfil}{.}{\rbrace} +\newcases{drcases}{\quad}{% + $\m@th\displaystyle{##}$\hfil}{$\m@th\displaystyle{##}$\hfil}{.}{\rbrace} +\newcases{drcases*}{\quad}{% + $\m@th\displaystyle{##}$\hfil}{{##}\hfil}{.}{\rbrace} +\newcases{cases*}{\quad}{% + $\m@th{##}$\hfil}{{##}\hfil}{\lbrace}{.} +\def\MT_matrix_begin:N #1{% + \hskip -\arraycolsep + \MH_let:NwN \@ifnextchar \MH_nospace_ifnextchar:Nnn + \array{*\c@MaxMatrixCols #1}} +\def\MT_matrix_end:{\endarray \hskip -\arraycolsep} +\MaybeMHPrecedingSpacesOff +\newenvironment{matrix*}[1][c] + {\MT_matrix_begin:N #1} + {\MT_matrix_end:} +\newenvironment{pmatrix*}[1][c] + {\left(\MT_matrix_begin:N #1} + {\MT_matrix_end:\right)} +\newenvironment{bmatrix*}[1][c] + {\left[\MT_matrix_begin:N #1} + {\MT_matrix_end:\right]} +\newenvironment{Bmatrix*}[1][c] + {\left\lbrace\MT_matrix_begin:N #1} + {\MT_matrix_end:\right\rbrace} +\newenvironment{vmatrix*}[1][c] + {\left\lvert\MT_matrix_begin:N #1} + {\MT_matrix_end:\right\rvert} +\newenvironment{Vmatrix*}[1][c] + {\left\lVert\MT_matrix_begin:N #1} + {\MT_matrix_end:\right\lVert} +\def\MT_smallmatrix_begin:N #1{% + \Let@\restore@math@cr\default@tag + \baselineskip6\ex@ \lineskip1.5\ex@ \lineskiplimit\lineskip + \csname MT_smallmatrix_#1_begin:\endcsname +} +\def\MT_smallmatrix_end:{\crcr\egroup\egroup\MT_smallmatrix_inner_space:} +\def\MT_smallmatrix_l_begin:{\null\MT_smallmatrix_inner_space:\vcenter\bgroup + \ialign\bgroup$\m@th\scriptstyle##$\hfil&&\thickspace + $\m@th\scriptstyle##$\hfil\crcr +} +\def\MT_smallmatrix_c_begin:{\null\MT_smallmatrix_inner_space:\vcenter\bgroup + \ialign\bgroup\hfil$\m@th\scriptstyle##$\hfil&&\thickspace\hfil + $\m@th\scriptstyle##$\hfil\crcr +} +\def\MT_smallmatrix_r_begin:{\null\MT_smallmatrix_inner_space:\vcenter\bgroup + \ialign\bgroup\hfil$\m@th\scriptstyle##$&&\thickspace\hfil + $\m@th\scriptstyle##$\crcr +} +\newenvironment{smallmatrix*}[1][\MT_smallmatrix_default_align:] + {\MT_smallmatrix_begin:N #1} + {\MT_smallmatrix_end:} +\def\MT_fenced_sm_generator:nnn #1#2#3{% + \@ifundefined{#1}{% + \newenvironment{#1} + {\@nameuse{#1hook}\mathopen{}\mathclose\bgroup\left#2\MT_smallmatrix_begin:N c}% + {\MT_smallmatrix_end:\aftergroup\egroup\right#3}% + }{}% + \@ifundefined{#1*}{% + \newenvironment{#1*}[1][\MT_smallmatrix_default_align:]% + {\@nameuse{#1hook}\mathopen{}\mathclose\bgroup\left#2\MT_smallmatrix_begin:N ##1}% + {\MT_smallmatrix_end:\aftergroup\egroup\right#3}% + }{}% +} +\MT_fenced_sm_generator:nnn{psmallmatrix}() +\MT_fenced_sm_generator:nnn{bsmallmatrix}[] +\MT_fenced_sm_generator:nnn{Bsmallmatrix}\lbrace\rbrace +\MT_fenced_sm_generator:nnn{vsmallmatrix}\lvert\rvert +\MT_fenced_sm_generator:nnn{Vsmallmatrix}\lVert\rVert +\define@key{\MT_options_name:} + {smallmatrix-align}{\def\MT_smallmatrix_default_align:{#1}} +\define@key{\MT_options_name:} + {smallmatrix-inner-space}{\def\MT_smallmatrix_inner_space:{#1}} +\setkeys{\MT_options_name:}{ + smallmatrix-align=c, + smallmatrix-inner-space=\, +} + +\MHPrecedingSpacesOn +\newcommand*\smashoperator[2][lr]{ + \def\MT_smop_use:NNNNN {\@nameuse{MT_smop_smash_#1:NNNNN}} + \toks@{#2} + \expandafter\MT_smop_get_args:wwwNnNn + \the\toks@\@nil\@nil\@nil\@nil\@nil\@nil\@@nil +} +\def\MT_smop_remove_nil_vi:N #1\@nil\@nil\@nil\@nil\@nil\@nil{#1} +\def\MT_smop_mathop:n {\mathop} +\def\MT_smop_limits: {\limits} +\MH_new_boolean:n {smop_one} +\MH_new_boolean:n {smop_two} +\def\MT_smop_get_args:wwwNnNn #1#2#3#4#5#6#7\@@nil{% + \begingroup + \def\MT_smop_arg_A: {#1} \def\MT_smop_arg_B: {#2} + \def\MT_smop_arg_C: {#3} \def\MT_smop_arg_D: {#4} + \def\MT_smop_arg_E: {#5} \def\MT_smop_arg_F: {#6} + \def\MT_smop_arg_G: {#7} + \if_meaning:NN \MT_smop_arg_A: \MT_smop_mathop:n + \if_meaning:NN \MT_smop_arg_C:\MT_smop_limits: + \def\MT_smop_final_arg_A:{#1{#2}}% + \if_meaning:NN \MT_smop_arg_D: \@nnil + \else: + \MH_set_boolean_T:n {smop_one} + \MH_let:NwN \MT_smop_final_arg_B: \MT_smop_arg_D: + \MH_let:NwN \MT_smop_final_arg_C: \MT_smop_arg_E: + \if_meaning:NN \MT_smop_arg_F: \@nnil + \else: + \MH_set_boolean_T:n {smop_two} + \MH_let:NwN \MT_smop_final_arg_D: \MT_smop_arg_F: + \edef\MT_smop_final_arg_E: + {\expandafter\MT_smop_remove_nil_vi:N \MT_smop_arg_G: } + \fi: + \fi: + \else: + \def\MT_smop_final_arg_A:{#1{#2}}% + \if_meaning:NN \MT_smop_arg_D: \@nnil + \else: + \MH_set_boolean_T:n {smop_one} + \MH_let:NwN \MT_smop_final_arg_B: \MT_smop_arg_C: + \MH_let:NwN \MT_smop_final_arg_C: \MT_smop_arg_D: + \if_meaning:NN \MT_smop_arg_F: \@nnil + \else: + \MH_set_boolean_T:n {smop_two} + \MH_let:NwN \MT_smop_final_arg_D: \MT_smop_arg_E: + \MH_let:NwN \MT_smop_final_arg_E: \MT_smop_arg_F: + \fi: + \fi: + \fi: + \else: + \if_meaning:NN \MT_smop_arg_B:\MT_smop_limits: + \def\MT_smop_final_arg_A:{#1}% + \if_meaning:NN \MT_smop_arg_D: \@nnil + \else: + \MH_set_boolean_T:n {smop_one} + \MH_let:NwN \MT_smop_final_arg_B: \MT_smop_arg_C: + \MH_let:NwN \MT_smop_final_arg_C: \MT_smop_arg_D: + \if_meaning:NN \MT_smop_arg_F: \@nnil + \else: + \MH_set_boolean_T:n {smop_two} + \MH_let:NwN \MT_smop_final_arg_D: \MT_smop_arg_E: + \MH_let:NwN \MT_smop_final_arg_E: \MT_smop_arg_F: + \fi: + \fi: + \else: + \def\MT_smop_final_arg_A:{#1}% + \if_meaning:NN \MT_smop_arg_C: \@nnil + \else: + \MH_set_boolean_T:n {smop_one} + \MH_let:NwN \MT_smop_final_arg_B: \MT_smop_arg_B: + \MH_let:NwN \MT_smop_final_arg_C: \MT_smop_arg_C: + \if_meaning:NN \MT_smop_arg_D: \@nnil + \else: + \MH_set_boolean_T:n {smop_two} + \MH_let:NwN \MT_smop_final_arg_D: \MT_smop_arg_D: + \MH_let:NwN \MT_smop_final_arg_E: \MT_smop_arg_E: + \fi: + \fi: + \fi: + \fi: + \MH_if_boolean:nT {smop_one}{ + \MT_smop_measure:NNNNN + \MT_smop_final_arg_A: \MT_smop_final_arg_B: \MT_smop_final_arg_C: + \MT_smop_final_arg_D: \MT_smop_final_arg_E: + } + \MT_smop_use:NNNNN + \MT_smop_final_arg_A: \MT_smop_final_arg_B: \MT_smop_final_arg_C: + \MT_smop_final_arg_D: \MT_smop_final_arg_E: + \endgroup +} +\def\MT_smop_needed_args:NNNNN #1#2#3#4#5{% + \displaystyle #1 + \MH_if_boolean:nT {smop_one}{ + \limits#2{\MT_cramped_clap_internal:Nn \scriptstyle{#3}} + \MH_if_boolean:nT {smop_two}{ + #4{\MT_cramped_clap_internal:Nn \scriptstyle{#5}} + } + } +} +\def\MT_smop_measure:NNNNN #1#2#3#4#5{% + \MH_let:NwN \MT_saved_mathclap:Nn \MT_cramped_clap_internal:Nn + \MH_let:NwN \MT_cramped_clap_internal:Nn \@secondoftwo + \sbox\z@{$\m@th\MT_smop_needed_args:NNNNN #1#2#3#4#5$} + \MH_let:NwN \MT_cramped_clap_internal:Nn \MT_saved_mathclap:Nn + \sbox\tw@{$\m@th\displaystyle#1$} + \@tempdima=.5\wd0 + \advance\@tempdima-.5\wd2 +} +\def\MT_smop_smash_l:NNNNN #1#2#3#4#5{ + \MT_smop_needed_args:NNNNN #1#2#3#4#5\kern\@tempdima +} +\def\MT_smop_smash_r:NNNNN #1#2#3#4#5{ + \kern\@tempdima\MT_smop_needed_args:NNNNN #1#2#3#4#5 +} +\def\MT_smop_smash_lr:NNNNN #1#2#3#4#5{ + \MT_smop_needed_args:NNNNN #1#2#3#4#5 +} +\def\MT_vphantom:Nn {\v@true\h@false\MT_internal_phantom:N} +\def\MT_hphantom:Nn {\v@false\h@true\MT_internal_phantom:N} +\def\MT_phantom:Nn {\v@true\h@true\MT_internal_phantom:N} +\def\MT_internal_phantom:N #1{ + \ifmmode + \expandafter\mathph@nt\expandafter#1 + \else + \expandafter\makeph@nt + \fi +} +\newcommand*\adjustlimits[6]{ + \sbox\z@{$\m@th \displaystyle #1$} + \sbox\tw@{$\m@th \displaystyle #4$} + \@tempdima=\dp\z@ \advance\@tempdima-\dp\tw@ + \if_dim:w \@tempdima>\z@ + \mathop{#1}\limits#2{#3} + \else: + \mathop{#1\MT_vphantom:Nn \displaystyle{#4}}\limits + #2{ + \def\finsm@sh{\ht\z@\z@ \box\z@} + \mathsm@sh\scriptstyle{\MT_cramped_internal:Nn \scriptstyle{#3}} + \MT_vphantom:Nn \scriptstyle + {\MT_cramped_internal:Nn \scriptstyle{#6}} + } + \fi: + \if_dim:w \@tempdima>\z@ + \mathop{#4\MT_vphantom:Nn \displaystyle{#1}}\limits + #5 + { + \MT_vphantom:Nn \scriptstyle + {\MT_cramped_internal:Nn \scriptstyle{#3}} + \def\finsm@sh{\ht\z@\z@ \box\z@} + \mathsm@sh\scriptstyle{\MT_cramped_internal:Nn \scriptstyle{#6}} + } + \else: + \mathop{#4}\limits#5{#6} + \fi: +} +\newcommand\SwapAboveDisplaySkip{% + \noalign{\vskip-\abovedisplayskip\vskip\abovedisplayshortskip} +} + +\newcommand\MoveEqLeft[1][2]{\kern #1em & \kern -#1em} +\newcommand\Aboxed[1]{\let\bgroup{\romannumeral-`}\@Aboxed#1&&\ENDDNE} +\def\@Aboxed#1\ENDDNE{% + \ifnum0=`{}\fi \setbox \z@ + \hbox{$\displaystyle#1{}\m@th$\kern\fboxsep \kern\fboxrule }% + \edef\@tempa {\kern \wd\z@ &\kern -\the\wd\z@ \fboxsep + \the\fboxsep \fboxrule \the\fboxrule }\@tempa \boxed {#1#2}% +} +\MHInternalSyntaxOff +\def\ArrowBetweenLines{\relax + \iffalse{\fi\ifnum0=`}\fi + \@ifstar{\ArrowBetweenLines@auxI{00}}{\ArrowBetweenLines@auxI{01}}} +\def\ArrowBetweenLines@auxI#1{% + \@ifnextchar[% + {\ArrowBetweenLines@auxII{#1}}% + {\ArrowBetweenLines@auxII{#1}[\Updownarrow]}} +\def\ArrowBetweenLines@auxII#1[#2]{% + \ifnum0=`{\fi \iffalse}\fi + \expandafter\in@\expandafter{\@currenvir}% + {alignedat,aligned,gathered}% + \ifin@ \else + \notag + \fi% + \\ + \noalign{\nobreak\vskip-\baselineskip\vskip-\lineskip}% + \noalign{\expandafter\in@\expandafter{\@currenvir}% + {alignedat,aligned,gathered}% + \ifin@ \else\notag\fi% + }% + \if#1 &&\quad #2\else #2\quad\fi + \\\noalign{\nobreak\vskip-\lineskip}} + +\MHInternalSyntaxOn +\newcommand\vdotswithin[1]{% + {\mathmakebox[\widthof{\ensuremath{{}#1{}}}][c]{{\vdots}}}} +\newlength\origjot +\setlength\origjot{\jot} +\newdimen\l_MT_shortvdotswithinadjustabove_dim +\newdimen\l_MT_shortvdotswithinadjustbelow_dim +\define@key{\MT_options_name:} + {shortvdotsadjustabove}{\setlength\l_MT_shortvdotswithinadjustabove_dim{#1}} +\define@key{\MT_options_name:} + {shortvdotsadjustbelow}{\setlength\l_MT_shortvdotswithinadjustbelow_dim{#1}} +\setkeys{\MT_options_name:}{ + shortvdotsadjustabove=2.15\origjot, + shortvdotsadjustbelow=\origjot +} +\def\shortvdotswithin{\relax + \@ifstar{\MT_svwi_aux:nn{00}}{\MT_svwi_aux:nn{01}}} +\def\MT_svwi_aux:nn #1#2{ + \MTFlushSpaceAbove + \if#1 \vdotswithin{#2}& \else &\vdotswithin{#2} \fi + \MTFlushSpaceBelow +} +\def\MT_remove_tag_unless_inner:n #1{% + \begingroup + \def\etb@tempa##1|#1|##2\MT@END{\endgroup + \ifx\@empty##2\@empty\notag\fi}% + \expandafter\etb@tempa\expandafter|alignedat|aligned|split|#1|\MT@END} + %| emacs +\newcommand\MTFlushSpaceAbove{ + \expandafter\MT_remove_tag_unless_inner:n\expandafter{\@currenvir} + \\ + \noalign{% + \nobreak\vskip-\baselineskip\vskip-\lineskip% + \vskip-\l_MT_shortvdotswithinadjustabove_dim + \vskip-\origjot + \vskip\jot + }% + \noalign{ + \expandafter\MT_remove_tag_unless_inner:n\expandafter{\@currenvir} + } +} +\newcommand\MTFlushSpaceBelow{ + \\\noalign{% + \nobreak\vskip-\lineskip + \vskip-\l_MT_shortvdotswithinadjustbelow_dim + \vskip-\origjot + \vskip\jot + } +} + +\def\MH_nrotarrow:NN #1#2{% + \setbox0=\hbox{$\m@th#1\uparrow$}\dimen0=\dp0 + \setbox0=\hbox{% + \reflectbox{\rotatebox[origin=c]{90}{$\m@th#1\mkern2.22mu #2$}}}% + \dp0=\dimen0 \box0 \mkern2.3965mu +} +\def\MH_nuparrow: {% + \mathrel{\mathpalette\MH_nrotarrow:NN\nrightarrow} } +\def\MH_ndownarrow: {% + \mathrel{\mathpalette\MH_nrotarrow:NN\nleftarrow} } +\AtBeginDocument{% + \RequirePackage{graphicx}% + \@ifundefined{nrightarrow}{% + \providecommand\nuparrow{% + \PackageError{mathtools}{\string\nuparrow\space~ is~ + constructed~ from~ \string\nrightarrow,~ which~ is~ not~ + provided.~ Please~ load~ the~ amssymb~ package~ or~ similar}{} + }}{ \providecommand\nuparrow{\MH_nuparrow:}} + \@ifundefined{nleftarrow}{% + \providecommand\ndownarrow{% + \PackageError{mathtools}{\string\ndownarrow\space~ is~ + constructed~ from~ \string\nleftarrow,~ which~ is~ not~ + provided.~ Please~ load~ the~ amssymb~ package~ or~ similar}{} + }}{ \providecommand\ndownarrow{\MH_ndownarrow:}} } +\def\MH_bigtimes_scaler:N #1{% + \vcenter{\hbox{#1$\m@th\mkern-2mu\times\mkern-2mu$}}} +\def\MH_bigtimes_inner: { + \mathchoice{\MH_bigtimes_scaler:N \huge} % display style + {\MH_bigtimes_scaler:N \LARGE} % text style + {\MH_bigtimes_scaler:N {}} % script style + {\MH_bigtimes_scaler:N \footnotesize} % script script style +} +\def\MH_csym_bigtimes: {\mathop{\MH_bigtimes_inner:}\displaylimits} +\AtBeginDocument{ + \providecommand\bigtimes{\MH_csym_bigtimes:} +} +\MH_let:NwN \MT_orig_intertext: \intertext@ +\newdimen\l_MT_above_intertext_sep +\newdimen\l_MT_below_intertext_sep +\define@key{\MT_options_name:} + {aboveintertextdim}{\setlength\l_MT_above_intertext_sep{#1}} +\define@key{\MT_options_name:} + {belowintertextdim}{\setlength\l_MT_below_intertext_sep{#1}} +\define@key{\MT_options_name:} + {above-intertext-dim}{\setlength\l_MT_above_intertext_sep{#1}} +\define@key{\MT_options_name:} + {below-intertext-dim}{\setlength\l_MT_below_intertext_sep{#1}} +\define@key{\MT_options_name:} + {above-intertext-sep}{\setlength\l_MT_above_intertext_sep{#1}} +\define@key{\MT_options_name:} + {below-intertext-sep}{\setlength\l_MT_below_intertext_sep{#1}} +\def\MT_intertext: {% + \def\intertext##1{% + \ifvmode\else\\\@empty\fi + \noalign{% + \penalty\postdisplaypenalty\vskip\belowdisplayskip + \vskip-\lineskiplimit % CCS + \vskip\normallineskiplimit % CCS + \vskip\l_MT_above_intertext_sep + \vbox{\normalbaselines + \ifdim + \ifdim\@totalleftmargin=\z@ + \linewidth + \else + -\maxdimen + \fi + =\columnwidth + \else \parshape\@ne \@totalleftmargin \linewidth + \fi + \noindent##1\par}% + \penalty\predisplaypenalty\vskip\abovedisplayskip% + \vskip-\lineskiplimit % CCS + \vskip\normallineskiplimit % CCS + \vskip\l_MT_above_intertext_sep + }% +}} +\def\MT_orig_intertext_true: { \MH_let:NwN \intertext@ \MT_orig_intertext: } +\def\MT_orig_intertext_false: { \MH_let:NwN \intertext@ \MT_intertext: } +\define@key{\MT_options_name:}{original-intertext}[true]{ + \@nameuse{MT_orig_intertext_#1:} +} +\setkeys{\MT_options_name:}{ + original-intertext=false +} +\def\MT_orig_shortintertext:n #1{% + \ifvmode\else\\\@empty\fi + \noalign{% + \penalty\postdisplaypenalty\vskip\abovedisplayshortskip + \vbox{\normalbaselines + \if_dim:w + \if_dim:w \@totalleftmargin=\z@ + \linewidth + \else: + -\maxdimen + \fi: + =\columnwidth + \else: + \parshape\@ne \@totalleftmargin \linewidth + \fi: + \noindent#1\par}% + \penalty\predisplaypenalty\vskip\abovedisplayshortskip% + }% +} +\newdimen\l_MT_above_shortintertext_sep +\newdimen\l_MT_below_shortintertext_sep +\define@key{\MT_options_name:} + {aboveshortintertextdim}{\setlength \l_MT_above_shortintertext_sep{#1}} +\define@key{\MT_options_name:} + {belowshortintertextdim}{\setlength \l_MT_below_shortintertext_sep{#1}} +\define@key{\MT_options_name:} + {above-short-intertext-dim}{\setlength \l_MT_above_shortintertext_sep{#1}} +\define@key{\MT_options_name:} + {below-short-intertext-dim}{\setlength \l_MT_below_shortintertext_sep{#1}} +\define@key{\MT_options_name:} + {above-short-intertext-sep}{\setlength \l_MT_above_shortintertext_sep{#1}} +\define@key{\MT_options_name:} + {below-short-intertext-sep}{\setlength \l_MT_below_shortintertext_sep{#1}} +\define@key{\MT_options_name:} + {above-shortintertext-sep}{\setlength \l_MT_above_shortintertext_sep{#1}} +\define@key{\MT_options_name:} + {below-shortintertext-sep}{\setlength \l_MT_below_shortintertext_sep{#1}} +\setkeys{\MT_options_name:}{ + aboveshortintertextdim=3pt, + belowshortintertextdim=3pt +} +\def\MT_shortintertext:n #1{% + \ifvmode\else\\\@empty\fi + \noalign{% + \penalty\postdisplaypenalty\vskip\abovedisplayshortskip + \vskip-\lineskiplimit + \vskip\normallineskiplimit + \vskip\l_MT_above_shortintertext_sep + \vbox{\normalbaselines + \if_dim:w + \if_dim:w \@totalleftmargin=\z@ + \linewidth + \else: + -\maxdimen + \fi: + =\columnwidth + \else: + \parshape\@ne \@totalleftmargin \linewidth + \fi: + \noindent#1\par}% + \penalty\predisplaypenalty\vskip\abovedisplayshortskip% + \vskip-\lineskiplimit + \vskip\normallineskiplimit + \vskip\l_MT_below_shortintertext_sep + }% +} +\def\MT_orig_shortintertext_true: { \MH_let:NwN \shortintertext \MT_orig_shortintertext:n } +\def\MT_orig_shortintertext_false: { \MH_let:NwN \shortintertext \MT_shortintertext:n } +\define@key{\MT_options_name:}{original-shortintertext}[true]{ + \@nameuse{MT_orig_shortintertext_#1:} +} +\setkeys{\MT_options_name:}{ + original-shortintertext=false +} +\providecommand*\clap[1]{\hb@xt@\z@{\hss#1\hss}} +\providecommand*\mathllap[1][\@empty]{ + \ifx\@empty#1\@empty + \expandafter \mathpalette \expandafter \MT_mathllap:Nn + \else + \expandafter \MT_mathllap:Nn \expandafter #1 + \fi +} +\providecommand*\mathrlap[1][\@empty]{ + \ifx\@empty#1\@empty + \expandafter \mathpalette \expandafter \MT_mathrlap:Nn + \else + \expandafter \MT_mathrlap:Nn \expandafter #1 + \fi +} +\providecommand*\mathclap[1][\@empty]{ + \ifx\@empty#1\@empty + \expandafter \mathpalette \expandafter \MT_mathclap:Nn + \else + \expandafter \MT_mathclap:Nn \expandafter #1 + \fi +} +\def\MT_mathllap:Nn #1#2{{}\llap{$\m@th#1{#2}$}} +\def\MT_mathrlap:Nn #1#2{{}\rlap{$\m@th#1{#2}$}} +\def\MT_mathclap:Nn #1#2{{}\clap{$\m@th#1{#2}$}} +\providecommand*\mathmbox{\mathpalette\MT_mathmbox:nn} +\def\MT_mathmbox:nn #1#2{\mbox{$\m@th#1#2$}} +\providecommand*\mathmakebox{ + \@ifnextchar[ \MT_mathmakebox_I:w + \mathmbox} +\def\MT_mathmakebox_I:w[#1]{% + \@ifnextchar[ {\MT_mathmakebox_II:w[#1]} + {\MT_mathmakebox_II:w[#1][c]}} +\def\MT_mathmakebox_II:w[#1][#2]{ + \mathpalette{\MT_mathmakebox_III:w[#1][#2]}} +\def\MT_mathmakebox_III:w[#1][#2]#3#4{% + \@begin@tempboxa\hbox{$\m@th#3#4$}% + \setlength\@tempdima{#1}% + \hbox{\hb@xt@\@tempdima{\csname bm@#2\endcsname}}% + \@end@tempboxa} +\def\mathsm@sh#1#2{% + \setbox\z@\hbox{$\m@th#1{#2}$}{}\finsm@sh} +\providecommand*\cramped[1][\@empty]{ + \ifx\@empty#1\@empty + \expandafter \mathpalette \expandafter \MT_cramped_internal:Nn + \else + \expandafter \MT_cramped_internal:Nn \expandafter #1 + \fi +} +\def\MT_cramped_internal:Nn #1#2{ + \sbox\z@{$\m@th#1\nulldelimiterspace=\z@\radical\z@{#2}$} + \ifx#1\displaystyle + \dimen@=\fontdimen8\textfont3 + \advance\dimen@ .25\fontdimen5\textfont2 + \else + \dimen@=1.25\fontdimen8 + \ifx#1\textstyle\textfont + \else + \ifx#1\scriptstyle + \scriptfont + \else + \scriptscriptfont + \fi + \fi + 3 + \fi + \advance\dimen@-\ht\z@ \ht\z@=-\dimen@ + \box\z@ +} +\providecommand*\crampedllap[1][\@empty]{ + \ifx\@empty#1\@empty + \expandafter \mathpalette \expandafter \MT_cramped_llap_internal:Nn + \else + \expandafter \MT_cramped_llap_internal:Nn \expandafter #1 + \fi +} +\def\MT_cramped_llap_internal:Nn #1#2{ + {}\llap{\MT_cramped_internal:Nn #1{#2}} +} +\providecommand*\crampedclap[1][\@empty]{ + \ifx\@empty#1\@empty + \expandafter \mathpalette \expandafter \MT_cramped_clap_internal:Nn + \else + \expandafter \MT_cramped_clap_internal:Nn \expandafter #1 + \fi +} +\def\MT_cramped_clap_internal:Nn #1#2{ + {}\clap{\MT_cramped_internal:Nn #1{#2}} +} +\providecommand*\crampedrlap[1][\@empty]{ + \ifx\@empty#1\@empty + \expandafter \mathpalette \expandafter \MT_cramped_rlap_internal:Nn + \else + \expandafter \MT_cramped_rlap_internal:Nn \expandafter #1 + \fi +} +\def\MT_cramped_rlap_internal:Nn #1#2{ + {}\rlap{\MT_cramped_internal:Nn #1{#2}} +} +\newcommand{\MT_prescript_inner:}[4]{ + \@mathmeasure\z@#4{\MT_prescript_sup:{#1}} + \@mathmeasure\tw@#4{\MT_prescript_sub:{#2}} + \if_dim:w \wd\tw@>\wd\z@ + \setbox\z@\hbox to\wd\tw@{\hfil\unhbox\z@} + \else: + \setbox\tw@\hbox to\wd\z@{\hfil\unhbox\tw@} + \fi: + \mathop{} + \mathopen{\vphantom{\MT_prescript_arg:{#3}}}^{\box\z@}\sb{\box\tw@} + \MT_prescript_arg:{#3} +} +\DeclareRobustCommand{\prescript}[3]{ + \mathchoice + {\MT_prescript_inner:{#1}{#2}{#3}{\scriptstyle}} + {\MT_prescript_inner:{#1}{#2}{#3}{\scriptstyle}} + {\MT_prescript_inner:{#1}{#2}{#3}{\scriptscriptstyle}} + {\MT_prescript_inner:{#1}{#2}{#3}{\scriptscriptstyle}} +} +\define@key{\MT_options_name:} + {prescript-sup-format}{\def\MT_prescript_sup:{#1}} +\define@key{\MT_options_name:} + {prescript-sub-format}{\def\MT_prescript_sub:{#1}} +\define@key{\MT_options_name:} + {prescript-arg-format}{\def\MT_prescript_arg:{#1}} +\setkeys{\MT_options_name:}{ + prescript-sup-format={}, + prescript-sub-format={}, + prescript-arg-format={}, +} +\ifx\e@alloc\@undefined% kernel thus older than 2015 + \def\@DeclareMathSizes #1#2#3#4#5{% + \@defaultunits\dimen@ #2pt\relax\@nnil + \if:w $#3$% + \MH_let:cN {S@\strip@pt\dimen@}\math@fontsfalse + \else: + \@defaultunits\dimen@ii #3pt\relax\@nnil + \@defaultunits\@tempdima #4pt\relax\@nnil + \@defaultunits\@tempdimb #5pt\relax\@nnil + \toks@{#1}% + \expandafter\xdef\csname S@\strip@pt\dimen@\endcsname{% + \gdef\noexpand\tf@size{\strip@pt\dimen@ii}% + \gdef\noexpand\sf@size{\strip@pt\@tempdima}% + \gdef\noexpand\ssf@size{\strip@pt\@tempdimb}% + \the\toks@ + }% + \fi: + } +\fi +\def\MT_mathic_true: { + \MH_if_boolean:nF {math_italic_corr}{ + \MH_set_boolean_T:n {math_italic_corr} + \MH_if_boolean:nTF {robustify}{ + \MH_let:NwN \MT_mathic_redeffer: \DeclareRobustCommand + }{ + \MH_let:NwN \MT_mathic_redeffer: \renewcommand + } + \MH_let:NwN \MT_begin_inlinemath: \( + %\renewcommand*\({ + \MT_mathic_redeffer:*\({ + \relax\ifmmode\@badmath\else + \ifhmode + \if_dim:w \fontdimen\@ne\font>\z@ + \if_dim:w \lastskip>\z@ + \skip@\lastskip\unskip + \if_num:w \lastpenalty>\z@ + \count@\lastpenalty\unpenalty + \fi: + \@@italiccorr + \if_num:w \count@>\z@ + \penalty\count@ + \fi: + \hskip\skip@ + \else: + \@@italiccorr + \fi: + \fi: + \fi: + $\fi: + } + } +} +\def\MT_mathic_false: { + \MH_if_boolean:nT {math_italic_corr}{ + \MH_set_boolean_F:n {math_italic_corr} + \MH_if_boolean:nTF {robustify}{ + \edef\({\MT_begin_inlinemath:}% + \forced_EQ_MakeRobust\(% + }{ + \MH_let:NwN \( \MT_begin_inlinemath: + } + } +} +\MH_new_boolean:n {math_italic_corr} +\define@key{\MT_options_name:}{mathic}[true]{ + \@ifundefined{MT_mathic_#1:} + { \MT_true_false_error: + \@nameuse{MT_mathic_false:} + } + { \@nameuse{MT_mathic_#1:} } +} +\newenvironment{spreadlines}[1]{ + \setlength{\jot}{#1} + \ignorespaces +}{ \ignorespacesafterend } +\MaybeMHPrecedingSpacesOff +\newenvironment{MT_gathered_env}[1][c]{% + \RIfM@\else + \nonmatherr@{\begin{\@currenvir}}% + \fi + \null\,% + \if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi \bgroup + \Let@ \chardef\dspbrk@context\@ne \restore@math@cr + \spread@equation + \ialign\bgroup + \MT_gathered_pre: + \strut@$\m@th\displaystyle##$ + \MT_gathered_post: + \crcr +}{% + \endaligned + \MT_gathered_env_end: +} +\MHPrecedingSpacesOn +\newcommand*\newgathered[4]{ + \newenvironment{#1} + { \def\MT_gathered_pre:{#2} + \def\MT_gathered_post:{#3} + \def\MT_gathered_env_end:{#4} + \MT_gathered_env + }{\endMT_gathered_env} +} +\newcommand*\renewgathered[4]{ + \renewenvironment{#1} + { \def\MT_gathered_pre:{#2} + \def\MT_gathered_post:{#3} + \def\MT_gathered_env_end:{#4} + \MT_gathered_env + }{\endMT_gathered_env} +} +\newgathered{lgathered}{}{\hfil}{} +\newgathered{rgathered}{\hfil}{}{} +\renewgathered{gathered}{\hfil}{\hfil}{} +\newcommand*\splitfrac[2]{% + \genfrac{}{}{0pt}{1}% + {\textstyle#1\quad\hfill}% + {\textstyle\hfill\quad\mathstrut#2}% +} +\newcommand*\splitdfrac[2]{% + \genfrac{}{}{0pt}{0}{#1\quad\hfill}{\hfill\quad\mathstrut #2}% +} +\MH_if_boolean:nT {fixamsmath}{ +\def\place@tag{% + \iftagsleft@ + \kern-\tagshift@ + \if@fleqn + \if_num:w \xatlevel@=\tw@ + \kern-\@mathmargin + \fi: + \fi: + \if:w 1\shift@tag\row@\relax + \rlap{\vbox{% + \normalbaselines + \boxz@ + \vbox to\lineht@{}% + \raise@tag + }}% + \else: + \rlap{\boxz@}% + \fi: + \kern\displaywidth@ + \else: + \kern-\tagshift@ + \if:w 1\shift@tag\row@\relax + \llap{\vtop{% + \raise@tag + \normalbaselines + \setbox\@ne\null + \dp\@ne\lineht@ + \box\@ne + \boxz@ + }}% + \else: + \llap{\boxz@}% + \fi: + \fi: +} +\def\x@calc@shift@lf{% + \if_dim:w \eqnshift@=\z@ + \global\eqnshift@\@mathmargin\relax + \alignsep@\displaywidth + \advance\alignsep@-\totwidth@ + \if_num:w \@tempcntb=0 + \else: + \global\divide\alignsep@\@tempcntb % original line + \fi: + \if_dim:w \alignsep@<\minalignsep\relax + \global\alignsep@\minalignsep\relax + \fi: + \fi: + \if_dim:w \tag@width\row@>\@tempdima + \saveshift@1% + \else: + \saveshift@0% + \fi:}% +} +\MaybeMHPrecedingSpacesOff +\renewcommand\aligned@a[1][c]{\start@aligned{#1}\m@ne} +\MHPrecedingSpacesOn +\endinput +%% +%% End of file `mathtools.sty'. diff --git a/books/mhsetup.sty b/books/mhsetup.sty new file mode 100644 index 0000000..b78969e --- /dev/null +++ b/books/mhsetup.sty @@ -0,0 +1,175 @@ +%% +%% This is file `mhsetup.sty', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mhsetup.dtx (with options: `package') +%% +%% This is a generated file. +%% +%% Copyright (C) 2002-2007,2010 by Morten Hoegholm +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either +%% version 1.3 of this license or (at your option) any later +%% version. The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of +%% LaTeX version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status "maintained". +%% +%% This Current Maintainer of this work is +%% Lars Madsen, Will Robertson and Joseph Wright. +%% +%% This work consists of the main source file mhsetup.dtx +%% and the derived files +%% mhsetup.sty, mhsetup.pdf, mhsetup.ins, mhsetup.drv. +%% +\ProvidesPackage{mhsetup}% + [2010/01/21 v1.2a programming setup (MH)] +\def\MHInternalSyntaxOn{ + \edef\MHInternalSyntaxOff{% + \catcode`\noexpand\~=\the\catcode`\~\relax + \catcode`\noexpand\ =\the\catcode`\ \relax + \catcode`\noexpand\^^I=\the\catcode`\^^I\relax + \catcode`\noexpand\@=\the\catcode`\@\relax + \catcode`\noexpand\:=\the\catcode`\:\relax + \catcode`\noexpand\_=\the\catcode`\_\relax + \endlinechar=\the\endlinechar\relax + }% + \catcode`\~=10\relax + \catcode`\ =9\relax + \catcode`\^^I=9\relax + \makeatletter + \catcode`\_=11\relax + \catcode`\:=11\relax + \endlinechar=` % + \relax +} +\MHInternalSyntaxOn +\AtEndOfPackage{\MHInternalSyntaxOff} +\let\MH_let:NwN \let +\def\MH_let:cN #1#2{ + \expandafter\MH_let:NwN \csname#1\endcsname#2} +\def\MH_let:cc #1#2{ + \expandafter\MH_let:NwN\csname#1\expandafter\endcsname + \csname#2\endcsname} +\def\MH_new_boolean:n #1{ + \expandafter\@ifdefinable\csname if_boolean_#1:\endcsname{ + \@namedef{boolean_#1_true:} + {\MH_let:cN{if_boolean_#1:}\iftrue} + \@namedef{boolean_#1_false:} + {\MH_let:cN{if_boolean_#1:}\iffalse} + \@nameuse{boolean_#1_false:}% + } +} +\def\MH_set_boolean_F:n #1{ \@nameuse{boolean_#1_false:} } +\def\MH_set_boolean_T:n #1{ \@nameuse{boolean_#1_true:} } +\def\MH_if_boolean:nTF #1{ + \@nameuse{if_boolean_#1:} + \expandafter\@firstoftwo + \else: + \expandafter\@secondoftwo + \fi: +} +\def\MH_if_boolean:nT #1{ + \@nameuse{if_boolean_#1:} + \expandafter\@firstofone + \else: + \expandafter\@gobble + \fi: +} +\def\MH_if_boolean:nF #1{ + \@nameuse{if_boolean_#1:} + \expandafter\@gobble + \else: + \expandafter\@firstofone + \fi: +} +\@ifundefined{if:w}{\MH_let:NwN \if:w =\if}{} +\@ifundefined{if_meaning:NN}{\MH_let:NwN \if_meaning:NN =\ifx}{} +\@ifundefined{else:}{\MH_let:NwN \else:=\else}{} +\@ifundefined{fi:}{\MH_let:NwN \fi:=\fi}{} +\AtBeginDocument{ + \@ifundefined{if_num:w}{\MH_let:NwN \if_num:w =\ifnum}{} + \@ifundefined{if_dim:w}{\MH_let:NwN \if_dim:w =\ifdim}{} + \@ifundefined{if_case:w}{\MH_let:NwN \if_case:w =\ifcase}{} +} +\@ifundefined{or:}{\MH_let:NwN \or:=\or}{} +\def\MH_cs_to_str:N {\expandafter\@gobble\string} +\@ifundefined{eTeXversion} + { + \MH_let:NwN \MH_protected:\relax + \def\MH_setlength:dn{\setlength} + \def\MH_addtolength:dn{\addtolength} + } + { + \MH_let:NwN \MH_protected:\protected + \def\MH_setlength:dn #1#2{#1=\dimexpr#2\relax\relax} + \def\MH_addtolength:dn #1#2{\advance#1 \dimexpr#2\relax\relax} + } + +\def\MH_keyval_alias_with_addon:nnnn #1#2#3#4{ + \@namedef{KV@#1@#2}{\@nameuse{KV@#1@#3}#4} + \@namedef{KV@#1@#2@default}{\@nameuse{KV@#1@#3@default}#4}} +\def\MH_keyval_alias:nnn #1#2#3{ + \MH_keyval_alias_with_addon:nnnn {#1}{#2}{#3}{}} +\def\MH_use_choice_i:nnnn #1#2#3#4{#1} +\def\MH_use_choice_ii:nnnn #1#2#3#4{#2} +\def\MH_use_choice_iii:nnnn #1#2#3#4{#3} +\def\MH_use_choice_iv:nnnn #1#2#3#4{#4} +\long\def\MH_nospace_ifnextchar:Nnn #1#2#3{ + \MH_let:NwN\reserved@d=~#1 + \def\reserved@a{#2} + \def\reserved@b{#3} + \futurelet\@let@token\MH_nospace_nextchar: +} +\def\MH_nospace_nextchar:{ + \if_meaning:NN \@let@token\reserved@d + \MH_let:NwN \reserved@b\reserved@a + \fi: + \reserved@b +} +\long\def\MH_nospace_testopt:nn #1#2{ + \MH_nospace_ifnextchar:Nnn[ + {#1} + {#1[{#2}]} +} +\def\MH_nospace_protected_testopt:n #1{ + \if_meaning:NN \protect\@typeset@protect + \expandafter\MH_nospace_testopt:nn + \else: + \@x@protect#1 + \fi: +} +\@ifundefined{kernel@ifnextchar} + {\MH_let:NwN \kernel@ifnextchar \@ifnextchar} + {} +\MH_let:NwN \MH_kernel_xargdef:nwwn \@xargdef +\long\def\MH_nospace_xargdef:nwwn #1[#2][#3]#4{ + \@ifdefinable#1{ + \expandafter\def\expandafter#1\expandafter{ + \expandafter + \MH_nospace_protected_testopt:n + \expandafter + #1 + \csname\string#1\endcsname + {#3}} + \expandafter\@yargdef + \csname\string#1\endcsname + \tw@ + {#2} + {#4}}} +\providecommand*\MHPrecedingSpacesOff{ + \MH_let:NwN \@xargdef \MH_nospace_xargdef:nwwn +} +\providecommand*\MHPrecedingSpacesOn{ + \MH_let:NwN \@xargdef \MH_kernel_xargdef:nwwn +} +\def \MH_group_align_safe_begin: {\iffalse{\fi\ifnum0=`}\fi} +\def \MH_group_align_safe_end: {\ifnum0=`{}\fi} +\endinput +%% +%% End of file `mhsetup.sty'. diff --git a/books/ps/v105hammfig1.eps b/books/ps/v105hammfig1.eps new file mode 100644 index 0000000..ee07f48 --- /dev/null +++ b/books/ps/v105hammfig1.eps @@ -0,0 +1,627 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: GIMP PostScript file plugin V 1.17 by Peter Kirchgessner +%%Title: v105hammfig1.eps +%%CreationDate: Sat Sep 17 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books/ps/v105hammfig1.eps for Hammarling chapter +20160918 tpd books/ps/v105hammfig2.eps for Hammarling chapter +20160918 tpd books/ps/v105hammfig3.eps for Hammarling chapter +20160918 tpd books/ps/v105hammfig4.eps for Hammarling chapter +20160918 tpd books/ps/v105hammfig5.eps for Hammarling chapter +20160918 tpd books/ps/v105hammfig6.eps for Hammarling chapter +20160918 tpd books/ps/v105hammfig7.eps for Hammarling chapter +20160918 tpd books/ps/v105hammtable2.eps for Hammarling chapter 20160911 tpd src/axiom-website/patches.html 20160911.01.tpd.patch 20160911 tpd books/bookvolbib add Hamm05, Quality Computed Solutions 20160910 tpd src/axiom-website/patches.html 20160910.01.tpd.patch diff --git a/patch b/patch index 7393d7a..f669520 100644 --- a/patch +++ b/patch @@ -1,8 +1,416 @@ -books/bookvolbib add Hamm05, Quality Computed Solutions +books/bookvol10.5 add Sven Hammarling chapter Goal: Axiom Literate Programming -Permission to quote article. +Note that adding the Hammarling equations forced the addition of +mathtools.sty... which forced a new version of \matrix... which +forced a rewrite of several books. Code rot strikes. Sigh. + +\index{Acton, F.S.} +\begin{chunk}{axiom.bib} +@book{Acto70, + author = "Acton, F.S.", + title = "Numerical Methods that (Usually) Work", + year = "1970", + publisher = "Harper and Row", + address = "New York, USA" +} + +\end{chunk} + +\index{Acton, F.S.} +\begin{chunk}{axiom.bib} +@book{Acto96, + author = "Acton, F.S.", + title = "Real Computing Made Real: Preventing Errors in Scientific + and Engineering Calculations", + year = "1996", + publisher = "Princeton University Press", + address = "Princeton, N.J. USA", + isbn = "0-691-03663-2" +} + +\end{chunk} + +\index{Alefeld, G.} +\index{Mayer, G.} +\begin{chunk}{axiom.bib} +@article{Alef00, + author = "Alefeld, G. and Mayer, G.", + title = "Interval analysis: Theory and applications", + journal = "J. Comput. Appl. Math.", + volume = "121", + pages = "421-464", + year = "2000" +} + +\end{chunk} + +\index{Anderson, E.} +\index{Bai, Z.} +\index{Bischof, S.} +\index{Blackford, S.} +\index{Demmel, J.} +\index{Dongarra, J. J.} +\index{DuCroz, J.} +\index{Greenbaum, A.} +\index{Hammarling, S.} +\index{McKenney, A.} +\index{Sorensen, D. C.} +\begin{chunk}{axiom.bib} +@book{Ande99, + author = "Anderson, E. and Bai, Z. and Bischof, S. and Blackford, S. and + Demmel, J. and Dongarra, J. J. and DuCroz, J. and Greenbaum, A. + and Hammarling, S. and McKenney, A. Sorensen, D. C.", + title = "LAPACK Users' Guide", + publisher = "SIAM", + year = "1999", + isbn = "0-89871-447-8", + url = "www.netlib.org/lapack/lug/" +} + +\end{chunk} + +\index{Bindel, D.} +\index{Demmel, J.} +\index{Kahan, W.} +\index{Marques, O.} +\begin{chunk}{axiom.bib} +@article{Bind02, + author = "Bindel, D. and Demmel, J. and Kahan, W. and Marques, O.", + title = On computing Givens rotations reliably and efficiently", + journal = "ACM Trans. Math. Software", + volume = "28", + pages = "206-238", + year = "2002" +} + +\end{chunk} + +\index{Blackford, L. S.} +\index{Cleary, A.} +\index{Demmel, J.} +\index{Dhillon, I.} +\index{Dongarra, J. J.} +\index{Hammarling, S.} +\index{Petitet, A.} +\index{Ren, H.} +\index{Stanley, K.} +\index{Whaley, R. C.} +\begin{chunk}{axiom.bib} +@article{Blac97, + author = "Blackford, L. S. and Cleary, A. and Demmel, J. and Dhillon, I. + and Dongarra, J. J. and Hammarling, S. and Petitet, A. and + Ren, H. and Stanley, K. and Whaley, R. C.", + title = "Practical experience in the numerical dangers of heterogeneous + computing", + journal = "ACM Trans. Math. Software", + volume = "23", + pages = "133-147", + year = "1997" +} + +\end{chunk} + +\index{Brankin, R. W.} +\index{Gladwell, I.} +\begin{chunk}{axiom.bib} +@article{Bran97, + author = "Brankin, R. W. and Gladwell, I.", + title = "rksuite\_90: Fortran 90 software for ordinary differential + equation initial-value problems", + journal = "ACM Trans. Math. Software", + volume = "23", + pages = "402-415", + year = "1997" +} + +\end{chunk} + +\index{Brankin, R. W.} +\index{Gladwell, I.} +\index{Shampine, L. F.} +\begin{chunk}{axiom.bib} +@techreport{Bran92, + author = "Brankin, R. W. and Gladwell, I. and Shampine, L. F.", + title = "RKSUITE: A suite of runge-kutta codes for the initial value + problem for ODEs", + year = "1992", + institution = "Southern Methodist University, Dept of Math.", + number = "Softreport 92-S1", + type = "Technical Report" +} + +\end{chunk} + +\index{Britton, J. L.} +\begin{chunk}{axiom.bib} +@book{Brit92, + author = "Britton, J. L.", + title = "Collected Works of A. M. Turing: Pure Mathematics", + publisher = "North-Holland", + years = "1992", + isbn = "0-444-88059-3" +} + +\end{chunk} + +\index{Chaitin-Chatelin, F.} +\index{Fraysse, V.} +\begin{chunk}{axiom.bib} +@book{Chai96, + author = "Chaitin-Chatelin, F. and Fraysse, V.", + title = "Lectures on Finite Precision Computations", + publisher = "SIAM", + year = "1996", + isbn = "0-89871-358-7" +} + +\end{chunk} + +\index{Chan, T. F.} +\index{Golub, G. H.} +\index{LeVeque, R. J.} +\begin{chunk}{axiom.bib} +@article{Chan83, + author = "Chan, T. F. and Golub, G. H. and LeVeque, R. J.", + title = "Algorithms for computing the sample variance: Analysis and + recommendations", + journal = "The American Statistician", + volume = "37", + pages = "242-247", + year = "1983" +} + +\end{chunk} + +\index{Cools, R.} +\index{Haegemans, A.} +\begin{chunk}{axiom.bib} +@article{Cool03, + author = "Cools, R. and Haegemans, A.", + title = "Algorithm 824: CUBPACK: A package for automatic cubature; + framework description", + journal = "ACM Trans. Math. Software", + volume = "29", + pages = "287-296", + year = "2003" +} + +\end{chunk} + +\index{Cox, M. G.} +\index{Dainton, M. P.} +\index{Harris, P. M.} +\begin{chunk}{axiom.bib} +@techreport{Coxx00, + author = "Cox, M. G. and Dainton, M. P. and Harris, P. M.", + title = "Testing spreadsheets and other packages used in metrology: + Testing functions for the calculation of standard deviation", + year = "2000", + institution = "National Physical Lab, Teddington, Middlesex UK", + type = "Technical Report", + number = "NPL Report CMSC07/00" +} + +\end{chunk} + +\index{Dodson, D. S.} +\begin{chunk}{axiom.bib} +@article{Dods83, + author = "Dodson, D. S.", + title = "Corrigendum: Remark on 'Algorithm 539: Basic Linear Algebra + Subroutines for FORTRAN usage", + journal = "ACM Trans. Math. Software", + volume = "9", + pages = "140", + year = "1983" +} + +\end{chunk} + +\index{Dodson, D. S.} +\index{Grimes, R. G.} +\begin{chunk}{axiom.bib} +@article{Dods82, + author = "Dodson, D. S. and Grimes, R. G.", + title = "Remark on algorithm 539: Basic Linear Algebra Subprograms for + Fortran usage", + journal = "ACM Trans. Math. Software", + volume = "8", + pages = "403-404", + year = "1982" +} + +\end{chunk} + +\index{Dongarra, J. J.} +\index{DuCroz, J.} +\index{Hammarling, S.} +\index{Hanson, R. J.} +\begin{chunk}{axiom.bib} +@article{Dong88, + author = "Dongarra, J. J. and DuCroz, J. and Hammarling, S. and + Hanson, R. J.", + title = "An extended set of FORTRAN Basic Linear Algebra Subprograms", + journal = "ACM Trans. Math. Software", + volume = "14", + pages = "1-32", + year = "1988" +} + +\end{chunk} + +\index{Dongarra, J.} +\index{DuCroz, J.} +\index{Duff, I. S.} +\index{Hammarling, S.} +\begin{chunk}{axiom.bib} +@article{Dong90, + author = "Dongarra, J. and DuCroz, J. and Duff, I. S. and Hammarling, S.", + title = "A set of Level 3 Basic Linear Algebra Subprograms", + journal = "ACM Trans. Math. Software", + volume = "16", + pages = "1-28", + year = "1990" +} + +\end{chunk} + +\index{Dubrulle, A. A.} +\begin{chunk}{axiom.bib} +@article{Dubr83, + author = "Dubrulle, A. A.", + title = "A class of numerical methods for the computation of Pythagorean + sums", + journal = "IBM J. Res. Develop.", + volume = "27", + number = "6", + pages = "582-589", + year = "1983" +} + +\end{chunk} + +\index{Einarsson, B.} +\begin{chunk}{axiom.bib} +@book{Eina05, + author = "Einarsson, B.", + title = "Accuracy and Reliability in Scientific Computing", + publisher = "SIAM", + year = "2005", + isbn = "0-89871-584-9", + url = "http://www.nsc.liu.se/wg25/book/" +} + +\end{chunk} + +\index{Forsythe, G. E.} +\begin{chunk}{axiom.bib} +@article{Fors70, + author = "Forsythe, G. E.", + title = "Pitfalls in computations, or why a math book isn't enough", + journal = "Amer. Math. Monthly", + volume = "9", + pages = "931-995", + year = "1970" +} + +\end{chunk} + +\index{Forsythe, G. E.} +\begin{chunk}{axiom.bib} +@incollection{Fors69, + author = "Forsythe, G. E.", + title = "What is a satisfactory quadratic equation solver", + booktitle = "Constructive Aspects of the Fundamental Theorem of Algebra", + pages = "53-61", + publisher = "Wiley", + year = "1969" +} + +\end{chunk} + +\index{Fox, L.} +\begin{chunk}{axiom.bib} +@article{Foxx71, + author = "Fox, L.", + title = "How to get meaningless answers in scientific computations (and + what to do about it)", + journal = "IMA Bulletin", + volume = "7", + pages = "296-302", + year = "1971" +} + +\end{chunk} + +\index{Givens, W.} +\begin{chunk}{axiom.bib} +@techreport{Give54, + author = "Givens, W.", + title = "Numerical computation of the characteristic values of a real + symmetric matrix", + year = "1954", + institution = "Oak Ridge National Laboratory", + type = "Technical Report", + number = "ORNL-1574" +} + +\end{chunk} + +\index{Golub, G.H.} +\begin{chunk}{axiom.bib} +@article{Golu65, + author = "Golub, G.H.", + title = "Numerical methods for solving linear least squares problems", + journal = "Numer. Math.", + volume = "7", + pages = "206-216", + year = "1965" +} + +\end{chunk} + +\index{Golub, Gene H.} +\index{Van Loan, Charles F.} +\begin{chunk}{axiom.bib} +@book{Golu89, + author = "Golub, Gene H. and Van Loan, Charles F.", + title = "Matrix Computations", + publisher = "Johns Hopkins University Press", + year = "1989", + isbn = "0-8018-3772-3" +} + +\end{chunk} + +\index{Golub, Gene H.} +\index{Van Loan, Charles F.} +\begin{chunk}{axiom.bib} +@book{Golu96, + author = "Golub, Gene H. and Van Loan, Charles F.", + title = "Matrix Computations", + publisher = "Johns Hopkins University Press", + isbn = "978-0-8018-5414-9", + year = "1996" +} + +\end{chunk} + +\index{Hammarling S.} +\begin{chunk}{axiom.bib} +@article{Hamm85, + author = "Hammarling S.", + title = " The Singular Value Decomposition in Multivariate Statistics", + journal = "ACM Signum Newsletter", + volume = "20", + number = "3", + pages = "2--25", + year = "1985" +} + +\end{chunk} \index{Hammarling, Sven} \begin{chunk}{axiom.bib} @@ -19,3 +427,498 @@ Permission to quote article. \end{chunk} +\index{Hargreaves, G.} +\begin{chunk}{axiom.bib} +@mastersthesis{Harg02, + author = "Hargreaves, G.", + title = "Interval analysis in MATLAB", + school = "University of Manchester, Dept. of Mathematics", + year = "2002" +} + +\end{chunk} + +\index{Higham, Nicholas J.} +\begin{chunk}{axiom.bib} +@book{High02, + author = "Higham, Nicholas J.", + title = "Accuracy and stability of numerical algorithms", + publisher = "SIAM", + isbn = "0-89871-521-0", + year = "2002" +} + +\end{chunk} + +\index{Higham, Nicholas J.} +\begin{chunk}{axiom.bib} +@article{High88, + author = "Higham, Nicholas J.", + title = "FORTRAN codes for estimating the one-norm of a real or complex + matrix, with applications to condition estimation", + journal = "ACM Trans. Math. Soft", + volume = "14", + number = "4", + pages = "381-396", + year = "1988" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@misc{IEEE85, + author = "IEEE", + title = "ANSI/IEEE Standard for Binary Floating Point Arithmetic: + Std 754-1985", + publisher = "IEEE Press", + year = "1985" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@misc{IEEE87, + author = "IEEE", + title = "ANSI/IEEE Standard for Radix Independent Floating Point Arithmetic: + Std 854-1987", + publisher = "IEEE Press", + year = "1987" +} + +\end{chunk} + +\index{Kn\"usel, L.} +\begin{chunk}{axiom.bib} +@article{Knus98, + author = {Kn\"usel, L.}, + title = "On the accuracy of statistical distributions in Microsoft + Excel 97", + journal = "Comput. Statist. Data Anal.", + volume = "26", + pages = "375-377", + year = "1998" +} + +\end{chunk} + +\index{Kreinovich, V.} +\begin{chunk}{axiom.bib} +@misc{Krei05, + author = "Kreinovich, V.", + title = "Interval cmoputations", + year = "2005", + url = "http://www.cs.utep.edu/interval-comp/" +} + +\end{chunk} + +\index{Lawson, C. L.} +\index{Hanson, R. J.} +\begin{chunk}{axiom.bib} +@book{Laws75, + author = "Lawson, C. L. and Hanson, R. J.", + title = "Solving Least Squares Problems", + publisher = "Prentice-Hall", + year = "1974" +} + +\end{chunk} + +\index{Lawson, C. L.} +\index{Hanson, R. J.} +\begin{chunk}{axiom.bib} +@book{Laws95, + author = "Lawson, C. L. and Hanson, R. J.", + title = "Solving Least Squares Problems", + publisher = "SIAM", + isbn = "0-89871-356-0", + year = "1995" +} + +\end{chunk} + +\index{Lawson, C. L.} +\index{Hanson, R. J.} +\index{Kincaid, D.} +\index{Krogh, F. T.} +\begin{chunk}{axiom.bib} +@article{Laws79, + author = "Lawson, C. L. and Hanson, R. J. and Kincaid, D. and Krogh, F. T.", + title = "Basic Linear Algebra Subprograms for FORTRAN usage", + journal = "ACM Trans. Math. Software", + volume = "5", + pages = "308-323", + year = "1979" +} + +\end{chunk} + +\index{Martin, R. S.} +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@article{Mart68, + author = "Martin, R. S. and Wilkinson, J. H.", + title = "Similarity reduction ofa general matrix to Hessenberg form", + journal = "Numer. Math.", + volume = "12", + pages = "349-368", + year = "1968" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@misc{Math05, + author = "MathWorks", + title = "MATLAB", + publisher = "The Mathworks, Inc.", + url = "http://www.mathworks.com" +} + +\end{chunk} + +\index{McCullough, B. D.} +\index{Wilson, B.} +\begin{chunk}{axiom.bib} +@article{Mccu02, + author = "McCullough, B. D. and Wilson, B.", + title = "On the accuracy of statistical procedures in Microsoft Excel + 2000 and Excel XP", + journal = "Comput. Statist. Data Anal.", + volume = "40", + pages = "713-721", + year = "2002" +} + +\end{chunk} + +\index{McCullough, B. D.} +\index{Wilson, B.} +\begin{chunk}{axiom.bib} +@article{Mccu99, + author = "McCullough, B. D. and Wilson, B.", + title = "On the accuracy of statistical procedures in Microsoft Excel 97", + journal = "Comput. Statist. Data Anal.", + volume = "31", + pages = "27-37", + year = "1999" +} + +\end{chunk} + +\index{Metcalf, M.} +\index{Reid, J. K.} +\begin{chunk}{axiom.bib} +@book{Metc96, + author = "Metcalf, M. and Reid, J. K.", + title = "Fortran 90/95 Explained", + publisher = "Oxford University Press", + year = "1996" +} + +\end{chunk} + +\index{Metcalf, M.} +\index{Reid, J. K.} +\index{Cohen, M.} +\begin{chunk}{axiom.bib} +@book{Metc04, + author = "Metcalf, M. and Reid, J. K. and Cohen, M.", + title = "Fortran 95/2003 Explained", + publisher = "Oxford University Press", + year = "2004", + isbn = "0-19-852693-8" +} + +\end{chunk} + +\index{Moler, C.} +\index{Morrison, D.} +\begin{chunk}{axiom.bib} +@article{Mole83, + author = "Moler, C. and Morrison, D.", + title = "Replacing square roots by Pythagorena sums", + journal = "IBM J. Res. Develop.", + volume = "27", + number = "6", + pages = "577-581", + year = "1983" +} + +\end{chunk} + +\index{Moore, R. E.} +\begin{chunk}{axiom.bib} +@books{Moor79, + author = "Moore, R. E.", + title = "methods and Applications of Interval Analysis", + publisher = "SIAM", + year = "1979" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@misc{NAGa05, + author = "Numerical Algorithms Group", + title = "The NAG Library", + url = "http://www.nag.co.uk/numeric", + year = "2005" +} + +\end{chunk} + +\begin{chunk}{axiom.bib} +@misc{NAGb05, + author = "Numerical Algorithms Group", + title = "The NAG Fortran Library Manual", + url = "http://www.nag.co.uk/numeric/fl/manual/html/FLlibrarymanual.asp", + year = "2005" +} + +\end{chunk} + +\index{Overton, M. L.} +\begin{chunk}{axiom.bib} +@book{Over01, + author = "Overton, M. L.", + title = "Numerical Computing with IEEE Floating Point Arithmetic", + publisher = "SIAM", + year = "2001", + isbn = "0-89871-482-6" +} + +\end{chunk} + +\index{Piessens, R.} +\index{de Doncker-Kapenga, E.}, +\index{\"Uberhuber, C. W.} +\index{Kahaner, D. K.} +\begin{chunk}{axiom.bib} +@book{Pies83, + author = {Piessens, R. and de Doncker-Kapenga, E. and \"Uberhuber, C. W. + and Kahaner, D. K.}, + title = "QUADPACK - A Subroutine Package for Automatic Integration", + publisher = "Springer-Verlag", + year = "1983" +} + +\end{chunk} + +\index{Priest, D. M.} +\begin{chunk}{axiom.bib} +@article{Prie04, + author = "Priest, D. M.", + title = "Efficient scaling for complex division", + journal = "ACM Trans. Math. Software", + volume = "30", + pages = "389-401", + year = "2004" +} + +\end{chunk} + +\index{Rump, S. M.} +\begin{chunk}{axiom.bib} +@InProceedings{Rump99, + author = "Rump, S. M.", + title = "INTLAB - INTerval LABoratory", + booktitle = "Developments in Reliable Computing", + pages = "77-104", + publisher = "Kluwer Academic", + year = "1999" +} + +\end{chunk} + +\index{Shampine, L. F.} +\index{Gladwell, I.} +\begin{chunk}{axiom.bib} +@InProceedings{Sham92, + author = "Shampine, L. F. and Gladwell, I.", + title = "The next generation of runge-kutta codes", + booktitle = "Computational Ordinary Differential Equations", + pages = "145-164", + publisher = "Oxford University Press", + year = "1992" +} + +\end{chunk} + +\index{Smith, R. L.} +\begin{chunk}{axiom.bib} +@article{Smit62, + author = "Smith, R. L.", + title = "Algorithm 116: Complex division", + journal = "Communs. Ass. comput. Mach.", + volume = "5", + pages = "435", + year = "1962" +} + +\end{chunk} + +\index{Stewart, G. W.} +\begin{chunk}{axiom.bib} +@book{Stew98, + author = "Stewart, G. W.", + title = "Matrix Algorithms: Basic Decompositions, volume I", + publisher = "SIAM", + year = "1998", + isbn = "0-89871-414-1" +} + +\end{chunk} + +\index{Stewart, G. W.} +\begin{chunk}{axiom.bib} +@article{Stew85, + author = "Stewart, G. W.", + title = "A note on complex division", + journal = "ACM Trans. Math. Software", + volume = "11", + pages = "238-241", + year = "1985" +} + +\end{chunk} + +\index{Stewart, G. W.} +\index{Sun, J.} +\begin{chunk}{axiom.bib} +@book{Stew90, + author = "Stewart, G. W. and Sun, J.", + title = "Matrix Perturbation Theory", + publisher = "Academic Press", + year = "1990" +} + +\end{chunk} + +\index{Turing, A. M.} +\begin{chunk}{axiom.bib} +@article{Turi48, + author = "Turing, A. M.", + title = "Rounding-off errors in matrix processes", + journal = "Q. J. Mech. Appl. Math.", + volume = "1", + pages = "287-308", + year = "1948" +} + +\end{chunk} + +\index{Vignes, J.} +\begin{chunk}{axiom.bib} +@article{Vign93, + author = "Vignes, J.", + title = "A stochastic arithmetic for reliable scientific computation", + jouirnal = "Math. and Comp. in Sim.", + volume = "25", + pages = "233-261", + year = "1993" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@book{Wilk63, + author = "Wilkinson, J. H.", + title = "Rounding Erroors in Algebraic Processes", + publisher = "HMSO", + series = "Notes on Applied Science, No. 32", + year = "1963" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@book{Wilk65, + author = "Wilkinson, J. H.", + title = "The Algebraic Eigenvalue Problem", + publisher = "Oxford University Press", + year = "1965" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@InProceedings{Wilk84, + author = "Wilkinson, J. H.", + title = "The perfidious polynomial", + booktitle = "Studies in Numerical Analysis", + volume = "24", + chapter = "1", + pages = "1-28", + year = "1984" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@article{Wilk86, + author = "Wilkinson, J. H.", + title = "Error analysis revisited", + journal = "IMA Bulletin", + volume = "22", + pages = "192-200", + year = "1986" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@article{Wilk61, + author = "Wilkinson, J. H.", + title = "Error analysis of diret methods of matrix inversion", + journal = "J. ACM", + volume = "8", + pages = "281-330", + year = "1961" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@article{Wilk85, + author = "Wilkinson, J. H.", + title = "The state of the art in error analysis", + journal = "NAG Newsletter", + volume = "2/85", + pages = "5-28", + year = "1985" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\begin{chunk}{axiom.bib} +@article{Wilk60, + author = "Wilkinson, J. H.", + title = "Error analysis of floating-point computation", + journal = "Numer. Math.", + volume = "2", + pages = "319-340", + year = "1960" +} + +\end{chunk} + +\index{Wilkinson, J. H.} +\index{Reinsch, C.} +\begin{chunk}{axiom.bib} +@book{Wilk71, + author = "Wilkinson, J. H.", + title = "Handbook for Automatic Computation, V2, Linear Algebra", + publisher = "Springer-Verlag", + year = "1971" +} + +\end{chunk} + diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 4cc2ee7..797814f 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -5556,6 +5556,8 @@ src/input/spadcall.input demonstrate Spad as a DSL over Lisp
books/bookvolbib add Hamm05, Quality Computed Solutions
20160911.01.tpd.patch books/bookvolbib add Hamm05, Quality Computed Solutions
+20160918.01.tpd.patch +books/bookvol10.5 add Sven Hammarling chapter
-- 1.7.5.4