{"push_id":1207252437,"size":1,"distinct_size":1,"ref":"refs/heads/master","head":"94083eebc838536c17ffc4ea668a36ea026ecf8f","before":"865a78a7026917d5a6fcdbfe1b98d32df6ac1983","commits":[{"sha":"94083eebc838536c17ffc4ea668a36ea026ecf8f","author":{"name":"Tim Daly","email":"4bd61968a14985739af99726f833ad8afe5905c4@axiom-developer.org"},"message":"books/bookvolbib Axiom Citations in the Literature\n\nGoal: Axiom Literate Programming\n\n\\index{Corless, Robert M.}\n\\index{Jeffrey, David J.}\n\\begin{chunk}{axiom.bib}\n@article{Corl98,\n  author = \"Corless, Robert M. and Jeffrey, David J.\",\n  title = \"Graphing Elementary Riemann Surfaces\",\n  journal = \"SIGSAM Bulletin\",\n  volume = \"32\",\n  number = \"1\",\n  pages = \"11-17\",\n  year = \"1998\",\n  paper = \"Corl98.djvu\",\n  abstract =\n    \"This paper discusses one of the prettiest pieces of elementary\n    mathematics or computer algebra, that we have ever had the pleasure\n    to learn. The tricks that we discuss here are certainly ``well-known''\n    (that is, in the literature), but we didn't know them until recently,\n    and none of our immediate colleagues knew them either. Therefore we\n    believe that it is useful to publicize them further. We hope that\n    you find these ideas as pleasant and useful as we do.\"\n}\n\n\\end{chunk}\n\n\\index{Corless, Robert M.}\n\\index{Jeffrey, David J.}\n\\index{Knuth, Donald E.}\n\\begin{chunk}{axiom.bib}\n@misc{Corl97,\n  author = \"Corless, Robert M. and Jeffrey, David J. and Knuth, Donald E.\",\n  title = \"A Sequence of Series for The Lambert W Function\",\n  year = \"1997\",\n  paper = \"Corl97.pdf\",\n  abstract =\n    \"We give a uniform treatment of several series expansions for the\n    Lambert $W$ function, leading to an infinite family of new series.\n    We also discuss standardization, complex branches, a family of\n    arbitrary-order iterative methods for computation of $W_i$, and\n    give a theorem showing how to correctly solve another simple and\n    frequently occurring nonlinear equation in terms of $W$ and the\n    unwinding number\"\n}\n\n\\end{chunk}\n\n\\index{Chow, Timothy Y.}\n\\begin{chunk}{axiom.bib}\n@article{Chow99,\n  author = \"Chow, Timothy Y.\",\n  title = \"What is a closed-form number?\",\n  journal = \"The American Mathematical Monthly\",\n  volume = \"106\",\n  number = \"5\",\n  pages = \"440-448\",\n  paper = \"Chowxx.pdf\",\n  year = \"1999\"\n}\n\n\\end{chunk}\n\n\\index{Hur, Namhyun}\n\\index{Davenport, James H.}\n\\begin{chunk}{axiom.bib}\n@inproceedings{Hurx00,\n  author = \"Hur, Namhyun and Davenport, James H.\",\n  title = \"An exact real algebraic arithmetic with equality determination\",\n  booktitle = \"Proc. ISSAC 2000\",\n  series = \"ISSAC '00\",\n  pages = \"169-174\",\n  year = \"2000\",\n  paper = \"Hurx00.djvu\",\n  abstract =\n    \"We describe a new arithmetic model for real algebraic numbers with\n    an exact equality determination. The model represents a real algebraic\n    number as a pair of an arbitrary precision numerical value and a\n    symbolic expression. For the numerical part we currently (another\n    representation could be used) use the dyadic exact real number and\n    for the symbolic part we use a square-free polynomial for the real\n    algebraic number. In this model we show that we can decide exactly\n    the equality of real algebraic numbers.\"\n}\n\n\\end{chunk}\n\n\\index{Langley, Simon}\n\\index{Richardson, Daniel}\n\\begin{chunk}{axiom.bib}\n@article{Lang02,\n  author = \"Langley, Simon and Richardson, Daniel\",\n  title = \"What can we do with a Solution?\",\n  journal = \"Electronic Notes in Theoretical Computer Science\",\n  volume = \"66\",\n  number = \"1\",\n  year = \"2002\",\n  url = \"http://www.elsevier.nl/locate/entcs/volume66.html\",\n  paper = \"Lang02.pdf\",\n  abstract =\n    \"If $S=0$ is a system of $n$ equations and unknowns over $\\mathbb{C}$\n    and $S(\\alpha)=0$ to what extent can we compute with the point $\\alpha$?\n    In particular, can we decide whether or not a polynomial expressions\n    in the components of $\\alpha$ with integral coefficients is zero?\n    This question is considered for both algebraic and elementary systems\n    of equations.\"\n}\n\n\\end{chunk}\n\n\\index{Wang, Paul S.}\n\\begin{chunk}{axiom.bib}\n@article{Wang74,\n  author = \"Wang, Paul S.\",\n  title = \"The Undecidability of the Existence of Zeros of Real Elementary\n           Functions\",\n  journal = \"J. ACM\",\n  volume = \"21\",\n  number = \"4\",\n  pages = \"586-589\",\n  year = \"1974\",\n  paper = \"Wang74.djvu\",\n  abstract =\n    \"From Richardson's undecidability results, it is shown that the predicate\n    ``there exists a real number $r$ such that $G(r)=0$'' is recursively\n    undecidable for $G(x)$ in a class of functions which involves polynomials\n    and the sine function. The deduction follows that the convergence of a\n    class of improper integrals is recursively undecidable.\"\n}\n\n\\end{chunk}\n\n\\index{Daly, Timothy}\n\\begin{chunk}{axiom.bib}\n@article{Daly02\n  author = \"Daly, Timothy\",\n  title = \"Axiom as open source\",\n  journal = \"SIGSAM Bulletin\",\n  volume = \"36\",\n  number = \"1\",\n  pages = \"28-28\",\n  month = \"March\",\n  year = \"2002\",\n  keywords = \"axiomref\"\n}\n\n\\end{chunk}\n\n\\index{Jenks, Richard D.}\n\\begin{chunk}{axiom.bib}\n@article{Jenk92,\n  author = \"Jenks, Richard D.\",\n  title = \"SCRATCHPAD\",\n  volume = \"??\",\n  number = \"24\",\n  pages = \"16-17\",\n  month = \"October\",\n  year = \"1972\",\n  keywords = \"axiomref\",\n  abstract =\n    \"The following SCRATCHPAD solution of Problem \\#2 was run on a 1280K\n    virtual machine under CP/CMS time sharing system on a System/360\n    model 67. The conversation below is a modification of a program\n    originally written by Yngve Sundblad, August 1972. The program uses\n    symmetrised formulae, saves certain intermediate results, but does not\n    eliminate numerical factors in denominators\"\n}\n\n\\end{chunk}\n\n\\index{Norman, Arthur C.}\n\\begin{chunk}{axiom.bib}\n@article{Norm75a,\n  author = \"Norman, Arthur C.\",\n  title = \"The SCRATCHPAD Power Series Package\",\n  journal = \"SIGSAM\",\n  volume = \"9\",\n  number = \"1\",\n  pages = \"12-20\",\n  year = \"1975\",\n  comment = \"IBM T.J. Watson Research RC4998\",\n  keywords = \"axiomref\"\n}\n\n\\end{chunk}\n\n\\index{Griesmer, James H.}\n\\index{Jenks, Richard D.}\n\\index{Yun, David Y.Y.}\n\\begin{chunk}{axiom.bib}\n@article{Grie75a,\n  author = \"Griesmer, James H. and Jenks, Richard D. and Yun, David Y.Y.\"\n  title = \"A SCRATCHPAD solution to problem \\#7\",\n  journal = \"SIGSAM\",\n  volume = \"9\",\n  number = \"3\",\n  pages = \"13-17\",\n  year = \"1975\"\n}\n\n\\end{chunk}\n\n\\index{Miller, Bruce R.}\n\\begin{chunk}{axiom.bib}\n@misc{Mill95,\n  author = \"Miller, Bruce R.\",\n  title = \"An expression formatter for MACSYMA\",\n  keywords = \"axiomref\",\n  year = \"1995\",\n  paper = \"Mill95.pdf\",\n  abstract =\n    \"A package for formatting algebraic expressions in MACSYA is described.\n    It provides facilities for user-directed hierarchical structuring of\n    expressions, as well as for directing simplifications to selected\n    subexpressions. It emphasizes a semantic rther than syntactic description\n    of the desired form. The package also provides utilities for obtaining\n    efficiently the coefficients of polynomials, trigonometric sums and\n    power series. Similar capabilities would be useful in other computer\n    algebra systems.\"\n}\n\n\\end{chunk}\n\n\\index{Griesmer, James H.}\n\\index{Jenks, Richard D.}\n\\index{Yun, David Y.Y.}\n\\begin{chunk}{axiom.bib}\n@article{Grie75b,\n  author = \"Griesmer, James H. and Jenks, Richard D. and Yun, David Y.Y.\"\n  title = \"A FORMAT statement in SCRATCHPAD\",\n  journal = \"SIGSAM\",\n  volume = \"9\",\n  number = \"3\",\n  pages = \"24-25\",\n  year = \"1975\",\n  keywords = \"axiomref\",\n  abstract =\n    \"Algebraic manipulation covers branches of software, particularly list\n    processing, mathematics, notably logic and number theory, and\n    applications largely in physics. The lectures will deal with all of these\n    to a varying extent.\n}\n\n\\end{chunk}\n\n\\index{Blair, Fred W.}\n\\index{Griesmer, James H.}\n\\index{Jenks, Richard D.}\n\\begin{chunk}{axiom.bib}\n@inproceedings{Blai70,\n  author = \"Blair, Fred W. and Griesmer, James H. and Jenks, Richard D.\",\n  title = \"An interactive facility for symbolic mathematics\",\n  booktitle = \"Proc. International Computing Symposium, Bonn, Germany\",\n  year = \"1970\",\n  pages = \"394-419\",\n  keywords = \"axiomref\",\n  abstract =\n    \"The SCRATCHPAD/1 system is designed to provide an interactive symbolic\n    coputational facility for the mathematician user. The system features\n    a user language designed to capture the style and succinctness of\n    mathematical notation, together with a facility for conveniently\n    introducing new notations into the language. A comprehensive system\n    library incorporates symbolic capabilities provided by such systems as\n    SIN, MATHLAB, and REDUCE.\"\n}\n\n\\end{chunk}\n\n\\index{Jenks, Richard D.}\n\\begin{chunk}{axiom.bib}\n@article{Jenk79a,\n  author = \"Jenks, Richard D.\",\n  title = \"SCRATCHPAD/360: reflections on a language design\",\n  journal = \"SIGSAM\",\n  volume = \"13\",\n  number = \"1\",\n  pages = \"16-26\",\n  year = \"1979\",\n  keywords = \"axiomref\",\n  comment = \"IBM Research RC 7405\",\n  abstract =\n    \"The key concepts of the SCRATCHPAD language are described, assessed,\n    and illustrated by an example. The language was originally intended as\n    an interactive problem solving language for symbolic mathematics.\n    Nevertheless, as this paper intends to show, it can be used as a\n    programming language as well.\"\n}\n\n\\end{chunk}\n\n\\index{Ng, Edward W.}\n\\begin{chunk}{axiom.bib}\n@techreport{Ngxx80,\n  author = \"Ng, Edward W.\",\n  title = \"Symbolic-Numeric Interface: A Review\",\n  type = \"technical report\",\n  number = \"NASA-CR-162690 HC A02/MF A01\"\n  institution = \"NASA Jet Propulsion Lab\",\n  url = \"http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19800008508.pdf\",\n  paper = \"Ngxx80.pdf\",\n  keywords = \"axiomref\",\n  abstract =\n    \"This is a survey of recent activities that either used or encouraged the\n    potential use of a combination of symbolic and numerical calculations.\n    Symbolic calculations here primarily refer to the computer processing of\n    procedures from classical algebra, analysis and calculus. Numerical\n    calculations refer to both numerical mathematics research and scientific\n    computation. This survey is inteded to point out a large number of problem\n    areas where a co-operation of symbolic and numeric methods is likely to\n    bear many fruits. These areas include such classical operations as\n    differentiation and integration, such diverse activities as function\n    approximations andqualitative analysis, and such contemporary topics as\n    finite element calculations and computational complexity. It is contended\n    that other less obvious topics such as the fast Fourier transform, linear\n    algebra, nonlinear analysis and error analysis would also benefti from a\n    synergistic approach advocated here.\"\n}\n\n\\end{chunk}\n\n\\index{Fateman, Richard J.}\n\\begin{chunk}{axiom.bib}\n@article{Fate01,\n  author = \"Fateman, Richard J.\",\n  title = \"A Review of Macsyma\",\n  journal = \"IEEE Trans. Knowl. Eng.\",\n  volume = \"1\",\n  number = \"1\",\n  year = \"2001\",\n  url = \"http://people.eecs.berkeley.edu/~fateman/papers/mac82b.pdf\",\n  paper = \"Fate01.pdf\",\n  keywords = \"axiomref\",\n  abstract =\n    \"We review the successes and failures of the Macsyma algebraic\n    manipulation system from the point of view of one of the original\n    contributors. We provide a retrospective examination of some of the\n    controversial ideas that worked, and some that did not. We consider\n    input/output, language semantics, data types, pattern matching,\n    knowledge-adjunction, mathematical semantics, the user community,\n    and software engineering. We also comment on the porting of this\n    system to a variety of computing systems, and possible future\n    directions for algebraic manipulation system-building.\"\n}\n\n\\end{chunk}\n\n\\index{Griesmer, James H.}\n\\index{Jenks, Richard D.}\n\\begin{chunk}{axiom.bib}\n@article{Grie74,\n  author = \"Griesmer, James H. and Jenks, Richard D.\",\n  title = \"A solution to problem \\#4\": the lie transform\",\n  journal = \"SIGSAM Bulletin\",\n  volume = \"8\",\n  number = \"4\",\n  pages = \"12-13\",\n  year = \"1974\",\n  keywords = \"axiomref\",\n  abstract =\n    \"The following SCRATCHPAD conversation for carrying out the Lie\n    Transform computation represents a slight modification of one written\n    by Dr. David Barton, when he was a summer visitor during 1972 at the\n    Watson Research Center.\"\n}\n\n\\end{chunk}\n\n\\begin{chunk}{axiom.bib}\n@misc{SIGS16,\n  author = \"SIGSAM, ACM\",\n  title = \"Axiom\"\n  url = \"http://www.sigsam.org/software/axiom.html\",\n  year = \"2016\",\n  contact = \"Infodir\\_SIGSAM\\@acm.org\",\n  abstract =\n    \"Axiom is a free, open source, general-purpose computer algebra\n    system. It features a strongly typed language. The system has an\n    interactive interpreter and a compiler. It includes over 1100\n    supported categories, domains, and packages covering large areas of\n    Mathematics.\"\n}\n\n\\end{chunk}\n\n\\index{Li, Yue}\n\\index{Dos Reis, Gabriel}\n\\begin{chunk}{axiom.bib}\n@inproceedings{Lixx11,\n  author = \"Li, Yue and Dos Reis, Gabriel\",\n  title = \"An Automatic Parallelization Framework for Algebraic\n           Computation Systems\",\n  booktitle = \"Proc. ISSAC 2011\",\n  pages = \"233-240\",\n  isbn = \"978-1-4503-0675-1\",\n  year = \"2011\",\n  url = \"http://www.axiomatics.org/~gdr/concurrency/oa-conc-issac11.pdf\",\n  paper = \"Lixx11.pdf\",\n  keywords = \"axiomref\",\n  abstract =\n    \"This paper proposes a non-intrusive automatic parallelization\n    framework for typeful and property-aware computer algebra systems.\n    Automatic parallelization remains a promising computer program\n    transformation for exploiting ubiquitous concurrency facilities\n    available in modern computers. The framework uses semantics-based\n    static analysis to extract reductions in library components based on\n    algebraic properties. An early implementation shows up to 5 times\n    speed-up for library functions and homotopy-based polynomial system\n    solver. The general framework is applicable to algebraic computation\n    systems and programming languages with advanced type systems that\n    support user-defined axioms or annotation systems.\"\n}\n\n\\end{chunk}\n\n\\index{Kendall, Wilfrid S.}\n\\begin{chunk}{axiom.bib}\n@article{Kend01,\n  author = \"Kendall, Wilfrid S.\",\n  title = \"Symbolic It\\^o calculus in AXIOM: an ongoing story\",\n  journal = \"Statistics and Computing\",\n  volume = \"11\",\n  pages = \"25-35\",\n  year = \"2001\",\n  url = \"http://www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/kendall/personal/ppt/327.ps.gz\",\n  paper = \"Kend01.pdf\",\n  keywords = \"axiomref\",\n  abstract =\n    \"Symbolic It\\^o calculus refers both to the implementation of the\n    It\\^o calculus algebra package and to its application. This article\n    reports on progress in the implementation of It\\^o calculus in the\n    powerful and innovative computer algebra package AXIOM, in the context\n    of a decade of previous implementations and applications. It is shown\n    how the elegant algebraic structure underylying the expressive and\n    effective formalism of It\\^o calculus can be implemented directly in\n    AXIOM using the package's programmable facilities for ``strong\n    typing'' of computational objects.  An application is given of the use\n    of the implementation to provide calculations for a new proof, based\n    on stochastic differentials, of the Mardia-Dryden distribution from\n    statistical shape theory.\"\n}\n\n\\end{chunk}\n\n\\index{Senechaud, Pascale}\n\\index{Siebert, F.}\n\\begin{chunk}{axiom.bib}\n@techreport{Sene87a,\n  author = \"Senechaud, Pascale and Siebert, F.\",\n  title = \"Etude dl l'algorithme de Kovacic et son implantation sur\n           Scratchpad II\",\n  type = \"Technical Report\",\n  number = \"639\",\n  institution = \"Institut IMAG, Informatique et Mathematiques Appliquees\n                 de Grenoble\",\n  address = \"Grenoble, France\",\n  year = \"1987\",\n  keywords = \"axiomref\"\n}\n\n\\end{chunk}\n\n\\index{Terelius, Bjorn}\n\\begin{chunk}{axiom.bib}\n@mastersthesis{Tere09,\n  author = \"Terelius, Bjorn\",\n  title = \"Symbolic Integration\",\n  school = \"Royal Institute of Technology\",\n  address = \"Stockholm, Sweden\",\n  year = \"2009\",\n  paper = \"Tere09.pdf\",\n  abstract =\n    \"Symbolic integration is the problem of expressing an indefinite integral\n    $\\int{f}$ of a given function $f$ as a finite combination $g$ of elementary\n    functions, or more generally, to determine whether a certain class of\n    functions contains an element $g$ such that $g^\\prime = f$.\"\n\n    In the first part of this thesis, we compare different algorithms for\n    symbolic integration. Specifically, we review the integration rules\n    taught in calculus courses and how they can be used systematically to\n    create a reasonable, but somewhat limited, integration method. Then we\n    present the differential algebra required to prove the transcendental\n    cases of Risch's algorithm. Risch's algorithm decides if the integral\n    of an elementary function is elementary and if so computes it. The\n    presentation is mostly self-contained and, we hope, simpler than\n    previous descriptions of the algorithm. Finally, we describe\n    Risch-Norman's algorithm which, although it is not a decision\n    procedure, works well in practice and is considerably simpler than the\n    full Risch algorithm.\n\n    In the second part of this thesis, we briefly discuss an\n    implementation of a computer algebra system and some of the\n    experiences it has given us.  We also demonstrate an implementation of\n    the rule-based approach and how it can be used, not only to compute\n    integrals, but also to generate readable derivations of the results.\"\n}\n\n\\end{chunk}\n\n\\index{Kendall, Wilfrid S.}\n\\begin{chunk}{axiom.bib}\n@article{Kend07,\n  author = \"Kendall, Wilfrid S.\",\n  title = \"Coupling all the Levy Stochastic Areas of Multidimensional\n           Brownian Motion\",\n  journal = \"The Annals of Probability\",\n  volume = \"35\",\n  number = \"3\",\n  pages = \"935-953\",\n  year = \"2007\",\n  keywords = \"axiomref\",\n  comment = \"Author used Axiom for computation but says missed citation\",\n  url = \"http://arxiv.org/pdf/math/0512336v2.pdf\",\n  paper = \"Kend07.pdf\",\n  abstract =\n    \"It is shown how to construct a successful co-adapted coupling of two\n    copies of an $n$-dimensional Brownian motion ($B_1,\\ldots,B_n$) while\n    simultaneously coupling all corresponding copies of the L{\\'e}vy\n    stochastic areas $\\int B_idB_j$ - \\int B_j dB_i$. It is conjectured\n    that successful co-adapted couplings still exist when the L{\\'e}vy\n    stochastic areas are replaced by a finite set of multiply iterated\n    path- and time-integrals, subject to algebraic compatibility of the\n    initial conditions.\"\n}\n\n\\end{chunk}\n\n\\index{Bronstein, Manuel}\n\\index{Lafaille, S\\'ebastien}\n\\begin{chunk}{axiom.bib}\n@inproceedings{Bron02,\n  author = \"Bronstein, Manuel and Lafaille, S\\'ebastien\",\n  title = \"Solutions of linear ordinary differential equations in terms\n           of special functions\",\n  booktitle = \"Proc. ISSAC '02\",\n  publisher = \"ACM Press\",\n  pages = \"23-28\",\n  year = \"2002\",\n  isbn = \"1-58113-484-3\",\n  url =\n   \"http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf\",\n  paper = \"Bron02.pdf\",\n  url2 = \"http://xena.hunter.cuny.edu/ksda/papers/bronstein2.pdf\",\n  paper2 = \"Bron02x.pdf\",\n  abstract =\n    \"We describe a new algorithm for computing special function solutions\n    of the form $y(x) = m(x)F(\\eta(x))$ of second order linear ordinary\n    differential equations, where $m(x)$ is an arbitrary Liouvillian\n    function, $\\eta(x)$ is an arbitrary rational function, and $F$\n    satisfies a given second order linear ordinary differential\n    equations. Our algorithm, which is base on finding an appropriate\n    point transformation between the equation defining $F$ and the one to\n    solve, is able to find all rational transformations for a large class\n    of functions $F$, in particular (but not only) the $_0F_1$ and $_1F_1$\n    special functions of mathematical physics, such as Airy, Bessel,\n    Kummer and Whittaker functions. It is also able to identify the values\n    of the parameters entering those special functions, and can be\n    generalized to equations of higher order.\"\n}\n\n\\end{chunk}\n\n\\index{Chan, L.}\n\\index{Cheb-Terrab, E.S.}\n\\begin{chunk}{axiom.bib}\n@inproceedings{Chan04,\n  author = \"Chan, L. and Cheb-Terrab, E.S.\",\n  title = \"Non-Liouvillian solutions for second order linear ODEs\",\n  booktitle = \"Proc. ISSAC 04\",\n  pages = \"80-86\",\n  isbn = \"1-58113-827-X\",\n  url = \"http://www.cecm.sfu.ca/CAG/papers/edgardoIS04.pdf\",\n  keywords = \"axiomref\",\n  paper = \"Chan04.pdf\",\n  abstract =\n    \"There exist sound literature and algorithms for computing Liouvillian\n    solutions for the important problem of linear ODEs with rational\n    coefficients. Taking as sample the 363 second order equations of that\n    type found in Kamke's book, for instance, 51\\% of them admit Liouvillian\n    solutions and so are solvable using Kovacic's algorithm. On the other\n    hand, special function solutions not admitting Liouvillian form appear\n    frequently in mathematical physics, but there are not so general\n    algorithms for computing them. In this paper we present an algorithm\n    for computing special function solutions which can be expressed using\n    the $_2F_1$, $_1F_1$ or $_0F_1$ hypergeometric functions. They algorithm\n    is easy to implement in the framework of a computer algebra system and\n    systematically solves 91\\% of the 363 Kamke's linear ODE examples\n    mentioned.\"\n}\n\n\\end{chunk}\n\n\\index{van Hoeij, Mark}\n\\index{Monagan, Michael}\n\\begin{chunk}{axiom.bib}\n@inproceedings{Hoei04,\n  author = \"van Hoeij, Mark and Monagan, Michael\",\n  title = \"Algorithms for Polynomial GCD Computation over Algebraic\n           Function Fields\",\n  booktitle = \"Proc. ISSAC 04\",\n  isbn = \"1-58113-827-X\",\n  url = \"http://www.cecm.sfu.ca/personal/mmonagan/papers/AFGCD.pdf\",\n  paper = \"Hoei04.pdf\",\n  abstract =\n    \"Let $L$ be an algebraic function field in $k \\ge 0$ parameters\n    $t_1,\\ldots,t)k$. Let $f_1$, $f_2$ be non-zero polynomials in\n    $L[x]$. We give two algorithms for computing their gcd. The first, a\n    modular GCD algorithm, is an extension of the modular GCD algorithm\n    for Brown for {\\bf Z}$[x_1,\\ldots,x_n]$ and Encarnacion for {\\bf\n    Q}$(\\alpha[x])$ to function fields. The second, a fraction-free\n    algorithm, is a modification of the Moreno Maza and Rioboo algorithm\n    for computing gcds over triangular sets. The modification reduces\n    coefficient grownth in $L$ to be linear.  We give an empirical\n    comparison of the two algorithms using implementations in Maple.\"\n}\n\n\\end{chunk}\n\n\\begin{chunk}{axiom.bib}\n@misc{Maxi16a,\n  author = \"Maxima\",\n  title = \"Symbolic Integration: The Algorithms\",\n  url =\n\"http://maxima.sourceforge.net/docs/tutorial/en/gaertner-tutorial-revision/Pages/SI001.htm\"\n}\n\n\\index{Gil, I.}\n\\begin{chunk}{axiom.bib}\n@inproceedings{Gilx92,\n  author = \"Gil, I.\",\n  title = \"Computation of the Jordan canonical form of a square matrix\n           (using the Axiom programming language)\",\n  booktitle = \"Proc ISSAC 1992\",\n  series = \"ISSAC '92\",\n  publisher = \"ACM\",\n  pages = \"138-145\",\n  isbn = \"0-89791-489-9 (soft cover), 0-89791-490-2 (hard cover)\",\n  keywords = \"axiomref\",\n  abstract =\n    \"Presents an algorithm for computing: the Jordan form of a square\n    matrix with coefficients in a field K using the computer algebra\n    system Axiom. This system presents the advantage of allowing generic\n    programming. That is to say, the algorithm can first be implemented\n    for matrices with rational coefficients and then generalized to\n    matrices with coefficients in any field.  Therefore the author\n    presents the general method which is essentially based on the use of\n    the Frobenius form of a matrix in order to compute its Jordan form;\n    and then restricts attention to matrices with rational\n    coefficients. On the one hand the author streamlines the algorithm\n    froben which computes the Frobenius form of a matrix, and on the other\n    she examines in some detail the transformation from the Frobenius form\n    to the Jordan form, and gives the so called algorithm Jordform. The\n    author studies in particular, the complexity of this algorithm and\n    proves that it is polynomial when the coefficients of the matrix are\n    rational. Finally the author gives some experiments and a conclusion.\"\n}\n\n\\end{chunk}\n\n\\index{InnerNormalBasisFieldFunctions}\n\\index{Stinson, D.R.}\n\\begin{chunk}{axiom.bib}\n@article{Stin90,\n  author = \"Stinson, D.R.\",\n  title = \"Some observations on parallel Algorithms for fast exponentiation\n           in $GF(2^n)$\",\n  journal = \"Siam J. Comp.\",\n  volume = \"19\",\n  number = \"4\",\n  pages = \"711-717\",\n  year = \"1990\",\n  paper = \"Stin90.pdf\",\n  algebra = \"\\newline\\refto{package INBFF InnerNormalBasisFieldFunctions}\",\n  abstract =\n    \"A normal basis represention in $GF(2^n)$ allows squaring to be\n    accomplished by a cyclic shift. Algorithms for multiplication in\n    $GF(2^n)$ using a normal basis have been studied by several\n    researchers. In this paper, algorithms for performing exponentiation\n    in $GF(2^n)$ using a normal basis, and how they can be speeded up by\n    using parallelization, are investigated.\"\n}\n\n\\end{chunk}\n\n\\index{FiniteFieldPolynomialPackage}\n\\index{Lenstra, H. W.}\n\\index{Schoof, R. J.}\n\\begin{chunk}{axiom.bib}\n@article{Lens87,\n  author = \"Lenstra, H. W. and Schoof, R. J.\",\n  title = \"Primitive Normal Bases for Finite Fields\",\n  journal = \"Mathematics of Computation\",\n  volume = \"48\",\n  number = \"177\",\n  year = \"1987\",\n  pages = \"217-231\",\n  url = \"http://www.math.leidenuniv.nl/~hwl/PUBLICATIONS/\",\n  paper = \"Lens87.pdf\",\n  algebra = \"\\newline\\refto{package FFPOLY FiniteFieldPolynomialPackage}\",\n  abstract =\n    \"It is proved that any finite extension of a finite field has a normal\n    basis consisting of primitive roots\"\n}\n\n\\end{chunk}\n\n\\index{CharacteristicNonZero}\n\\index{FieldOfPrimeCharacteristic}\n\\index{ExtensionField}\n\\index{FiniteFieldCategory}\n\\index{FiniteAlgebraicExtensionField}\n\\index{SimpleAlgebraicExtension}\n\\index{InnerPrimeField}\n\\index{PrimeField}\n\\index{FiniteFieldExtensionByPolynomial}\n\\index{FiniteFieldCyclicGroupExtensionByPolynomial}\n\\index{FiniteFieldNormalBasisExtensionByPolynomial}\n\\index{FiniteFieldExtension}\n\\index{FiniteFieldCyclicGroupExtension}\n\\index{FiniteFieldNormalBasisExtension}\n\\index{InnerFiniteField}\n\\index{FiniteField}\n\\index{FiniteFieldCyclicGroup}\n\\index{FiniteFieldNormalBasis}\n\\index{DiscreteLogarithmPackage}\n\\index{FiniteFieldFunctions}\n\\index{InnerNormalBasisFieldFunctions}\n\\index{FiniteFieldPolynomialPackage}\n\\index{FiniteFieldPolynomialPackage2}\n\\index{FiniteFieldHomomorphisms}\n\\index{FiniteFieldFactorizationWithSizeParseBySideEffect}\n\\index{Grabmeier, Johannes}\n\\index{Scheerhorn, Alfred}\n\\begin{chunk}{axiom.bib}\n@techreport{Grab92,\n  author = \"Grabmeier, Johannes and Scheerhorn, Alfred\",\n  title = \"Finite fields in Axiom\",\n  type = \"technical report\",\n  number = \"AXIOM Technical Report TR7/92 (ATR/5)(NP2522)\",\n  institution = \"Numerical Algorithms Group, Inc.\",\n  address = \"Downer's Grove, IL, USA and Oxford, UK\",\n  year = \"1992\",\n  url = \"http://www.nag.co.uk/doc/TechRep/axiomtr.html\",\n  keywords = \"axiomref\",\n  paper = \"Grab92.pdf\",\n  algebra =\n   \"\\newline\\refto{category CHARNZ CharacteristicNonZero}\n    \\newline\\refto{category FPC FieldOfPrimeCharacteristic}\n    \\newline\\refto{category XF ExtensionField}\n    \\newline\\refto{category FFIELDC FiniteFieldCategory}\n    \\newline\\refto{category FAXF FiniteAlgebraicExtensionField}\n    \\newline\\refto{domain SAE SimpleAlgebraicExtension}\n    \\newline\\refto{domain IPF InnerPrimeField}\n    \\newline\\refto{domain PF PrimeField}\n    \\newline\\refto{domain FFP FiniteFieldExtensionByPolynomial}\n    \\newline\\refto{domain FFCGP FiniteFieldCyclicGroupExtensionByPolynomial}\n    \\newline\\refto{domain FFNBP FiniteFieldNormalBasisExtensionByPolynomial}\n    \\newline\\refto{domain FFX FiniteFieldExtension}\n    \\newline\\refto{domain FFCGX FiniteFieldCyclicGroupExtension}\n    \\newline\\refto{domain FFNBX FiniteFieldNormalBasisExtension}\n    \\newline\\refto{domain IFF InnerFiniteField}\n    \\newline\\refto{domain FF FiniteField}\n    \\newline\\refto{domain FFCG FiniteFieldCyclicGroup}\n    \\newline\\refto{domain FFNB FiniteFieldNormalBasis}\n    \\newline\\refto{package DLP DiscreteLogarithmPackage}\n    \\newline\\refto{package FFF FiniteFieldFunctions}\n    \\newline\\refto{package INBFF InnerNormalBasisFieldFunctions}\n    \\newline\\refto{package FFPOLY FiniteFieldPolynomialPackage}\n    \\newline\\refto{package FFPOLY2 FiniteFieldPolynomialPackage2}\n    \\newline\\refto{package FFHOM FiniteFieldHomomorphisms}\n    \\newline\\refto\n        {package FFFACTSE FiniteFieldFactorizationWithSizeParseBySideEffect}\",\n  abstract =\n    \"Finite fields play an important role for many applications (e.g. coding\n    theory, cryptograpy). There are different ways to construct a finite\n    field for a given prime power. The paper describes the different\n    constructions implemented in AXIOM. These are {\\sl polynomial basis\n    representation}, {\\sl cyclic group representation}, and {\\sl normal\n    basis representation}. Furthermore, the concept of the implementation,\n    the used algorithms and the various datatype coercions between these\n    representations are discussed.\"\n}\n\n\\end{chunk}\n\n\\index{InnerNormalBasisFieldFunctions}\n\\index{Itoh, T.}\n\\index{Tsujii, S.}\n\\begin{chunk}{axiom.bib}\n@article{Itoh88,\n  author = \"Itoh, T. and Tsujii, S.\",\n  title = \"A fast algorithm for computing multiplicative inverses in\n           $GF(2^m)$ using normal bases\",\n  journal = \"Inf. and Comp.\",\n  volume = \"78\",\n  pages = \"171-177\",\n  year = \"1988\",\n  paper = \"Itoh88.pdf\",\n  algebra = \"\\newline\\refto{package INBFF InnerNormalBasisFieldFunctions}\",\n  abstract =\n    \"This paper proposes a fast algorithm for computing multiplicative\n    inverses in $GF(2^m)$ using normal bases. Normal bases have the\n    following useful property: In the case that an element $x$ in\n    $GF(2^m)$ is represented by normal bases, $2^k$ power operation of an\n    element $x$ in $GF(2^m)$ can be carried out by $k$ times cyclic shift\n    of its vector representation.  C.C. Wang et al. proposed an algorithm\n    for computing multiplicative inverses using normal bases, which\n    requires $(m-2)$ multiplications in $GF(2^m)$ and $(m-1)$ cyclic\n    shifts. The fast algorithm proposed in this paper also uses normal\n    bases, and computes multiplicative inverses iterating multiplications\n    in $GF(2^m)$. It requires at most $2[log_2(m-1)]$ multiplications in\n    $GF(2^m)$ and $(m-1)$ cyclic shifts, which are much less than those\n    required in Wang's method. The same idea of the proposed fast\n    algorithm is applicable to the general power operation in $GF(2^m)$\n    and the computation of multiplicative inverses in $GF(q^m)$\n    $(q=2^n)$.\"\n}\n\n\\end{chunk}","distinct":true,"url":"https://api.github.com/repos/daly/axiom/commits/94083eebc838536c17ffc4ea668a36ea026ecf8f"}]}
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