Autour de l'évaluation numérique des fonctions D-finies

Les fonctions D-finies (ou holonomes) a une variable sont les solutions d'equations differentielles lineaires a coefficients polynomiaux. En calcul formel, il s'est avere fructueux depuis une vingtaine d'annees d'en developper un traitement algorithmique unifie. Cette these s'inscrit dans cette optique, et s'interesse a l'evaluation numerique des fonctions D-finies ainsi qu'a quelques problemes apparentes. Elle explore trois grandes directions. La premiere concerne la majoration des coefficients des developpements en serie de fonctions D-finies. On aboutit a un algorithme de calcul automatique de majorants accompagne d'un resultat de finesse des bornes obtenues. Une seconde direction est la mise en pratique de l'algorithme " bit burst " de Chudnovsky et Chudnovsky pour le prolongement analytique numerique a precision arbitraire des fonctions D-finies. Son implementation est l'occasion de diverses ameliorations techniques. Ici comme pour le calcul de bornes, on s'attache par ailleurs a couvrir le cas des points singuliers reguliers des equations differentielles. Enfin, la derniere partie de la these developpe une methode pour calculer une approximation polynomiale de degre impose d'une fonction D-finie sur un intervalle, via l'etude des developpements en serie de Tchebycheff de ces fonctions. Toutes les questions sont abordees avec un triple objectif de rigueur (resultats numeriques garantis), de generalite (traiter toute la classe des fonctions D-finies) et d'efficacite. Pratiquement tous les algorithmes etudies s'accompagnent d'implementations disponibles publiquement.

[1]  Jet Wimp,et al.  Computation with recurrence relations , 1986 .

[2]  Ekatharine A. Karatsuba,et al.  Fast evaluation of hypergeometric functions by FEE , 1999 .

[3]  Marius van der Put,et al.  Galois Theory of Difference Equations , 1997 .

[4]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[5]  F. Fauvet,et al.  Remarques algorithmiques liées au rang d’un opérateur différentiel linéaire , 2003 .

[6]  Frédéric Chyzak,et al.  An extension of Zeilberger's fast algorithm to general holonomic functions , 2000, Discret. Math..

[7]  George F. Corliss,et al.  Solving Ordinary Differential Equations Using Taylor Series , 1982, TOMS.

[8]  Nicolas Brisebarre,et al.  Chebyshev interpolation polynomial-based tools for rigorous computing , 2010, ISSAC.

[9]  Luc Rebillard.  Etude théorique et algorithmique des séries de Chebyshev solutions d'équations différentielles holonomes , 1998 .

[10]  Bruno Salvy,et al.  Chebyshev expansions for solutions of linear differential equations , 2009, ISSAC '09.

[11]  Howard Cheng,et al.  Time-and space-efficient evaluation of some hypergeometric constants , 2007, ISSAC '07.

[12]  Fabrice Bellard Jan.  Computation of 2700 billion decimal digits of Pi using a Desktop Computer , 2010 .

[13]  Markus Neher,et al.  An enclosure method for the solution of linear odes with polynomial coefficients , 1999 .

[14]  Yurii Lyubarskii,et al.  Lectures on entire functions , 1996 .

[15]  A. Krall.  Applied Analysis , 1986 .

[16]  Jeremy J. Gray,et al.  Linear Differential Equations and Group Theory from Riemann to Poincare , 1986 .

[17]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

[18]  Javier Sesma,et al.  An algorithm to obtain global solutions of the double confluent Heun equation , 2008, Numerical Algorithms.

[19]  Bruno Salvy,et al.  Examples of automatic asymptotic expansions , 1991, SIGS.

[20]  Mark van Hoeij,et al.  Formal Solutions and Factorization of Differential Operators with Power Series Coefficients , 1997, J. Symb. Comput..

[21]  L. Fox,et al.  Chebyshev Methods for Ordinary Differential Equations , 1962, Computer/law journal.

[22]  Victor Y. Pan,et al.  New fast algorithms for polynomial interpolation and evaluation on the Chebyshev node set , 1998 .

[23]  Harrison Tsai,et al.  Weyl Closure of a Linear Differential Operator , 2000, J. Symb. Comput..

[24]  Willard L. Miranker,et al.  Self-Validating Numerics for Function Space Problems , 1984 .

[25]  E. G. C. Poole,et al.  Introduction to the theory of linear differential equations , 1936 .

[26]  D. V. Chudnovsky,et al.  On expansion of algebraic functions in power and Puiseux series, I , 1986, J. Complex..

[27]  Alexis Darrasse,et al.  The Dynamic Dictionary of Mathematical Functions (DDMF) , 2010, ICMS.

[28]  Bruno Salvy,et al.  ESF: an automatically generated encyclopedia of special functions , 2003, ISSAC '03.

[29]  C. W. Clenshaw.  A note on the summation of Chebyshev series , 1955 .

[30]  F. Schäfke,et al.  Lösungstypen von Differenzengleichungen und Summengleichungen in normierten abelschen Gruppen , 1965 .

[31]  E. Salamin,et al.  Computation of π Using Arithmetic-Geometric Mean , 1976 .

[32]  V. Blondel,et al.  The presence of a zero in an integer linear recurrent sequence is NP-hard to decide , 2002 .

[33]  Victor Y. Pan,et al.  Acceleration of Euclidean Algorithm and Rational Number Reconstruction , 2003, SIAM J. Comput..

[34]  Cyril Chabaud.  Séries génératrices algébriques : asymptotique et applications combinatoires , 2002 .

[35]  E. N.,et al.  The Calculus of Finite Differences , 1934, Nature.

[36]  Markus Neher,et al.  Validated Bounds for Taylor Coefficients of Analytic Functions , 2001, TOMS.

[37]  Leonard Lipshitz,et al.  Decision problems for differential equations , 1989, Journal of Symbolic Logic.

[38]  S. Lang.  The Riemann Surface , 1972 .

[39]  J. Harrison,et al.  Efficient and accurate computation of upper bounds of approximation errors , 2011, Theor. Comput. Sci..

[40]  Erich Kaltofen,et al.  On fast multiplication of polynomials over arbitrary algebras , 1991, Acta Informatica.

[41]  Bruno Salvy,et al.  Effective asymptotics of linear recurrences with rational coefficients , 1996, Discret. Math..

[42]  Paul Kreuser,et al.  Über das Verhalten der Integrale homogener linearer Differenzengleichungen im Unendlichen , 1914 .

[43]  Richard J. Fateman,et al.  A MACSYMA package for the generation and manipulation of chebyshev series , 1989, ISSAC '89.

[44]  Richard P. Brent,et al.  The complexity of multiple-precision arithmetic , 2010, ArXiv.

[45]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

[46]  Henri Poincaré,et al.  Sur les Equations Lineaires aux Differentielles Ordinaires et aux Differences Finies , 1885 .

[47]  C. Lanczos,et al.  Trigonometric Interpolation of Empirical and Analytical Functions , 1938 .

[48]  Abraham Waksman.  On Winograd's Algorithm for Inner Products , 1970, IEEE Transactions on Computers.

[49]  Gerald Estrin,et al.  Organization of computer systems: the fixed plus variable structure computer , 1960, IRE-AIEE-ACM '60 (Western).

[50]  Bruno Haible,et al.  Fast Multiprecision Evaluation of Series of Rational Numbers , 1998, ANTS.

[51]  Stanisław Lewanowicz,et al.  A new approach to the problem of constructing recurrence relations for the Jacobi coefficients , 1991 .

[52]  T. J. Rivlin,et al.  Ultra-arithmetic II: intervals of polynomials , 1982 .

[53]  Bruno Salvy,et al.  GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable , 1994, TOMS.

[54]  V. Pan.  Optimal and nearly optimal algorithms for approximating polynomial zeros , 1996 .

[55]  Bruno Salvy,et al.  Effective bounds for P-recursive sequences , 2009, J. Symb. Comput..

[56]  M. Neher.  Improved validated bounds for Taylor coefficients and for Taylor remainder series , 2003 .

[57]  Richard P. Brent,et al.  Fast Multiple-Precision Evaluation of Elementary Functions , 1976, JACM.

[58]  Philippe Flajolet,et al.  Computer Algebra Libraries for Combinatorial Structures , 1995, J. Symb. Comput..

[59]  Philippe Flajolet,et al.  Singularity Analysis of Generating Functions , 1990, SIAM J. Discret. Math..

[60]  Manuel Bronstein,et al.  Full partial fraction decomposition of rational functions , 1993, ISSAC '93.

[61]  D. V. Chudnovsky,et al.  Computer assisted number theory with applications , 1987 .

[62]  C. W. Clenshaw.  The numerical solution of linear differential equations in Chebyshev series , 1957, Mathematical Proceedings of the Cambridge Philosophical Society.

[63]  Geertrui Klara Immink,et al.  Asymptotics of Analytic Difference Equations , 1984 .

[64]  Christoph Quirin Lauter,et al.  Sollya: An Environment for the Development of Numerical Codes , 2010, ICMS.

[65]  Steven C. Leth,et al.  An asymptotic formula for powers of binomial coefficients , 2004, The Mathematical Gazette.

[66]  Christoph Quirin Lauter.  Arrondi correct de fonctions mathématiques : fonctions univariées et bivariées, certification et automatisation , 2008 .

[67]  Tobin A. Driscoll,et al.  The chebop system for automatic solution of differential equations , 2008 .

[68]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[69]  Between Decidability.  Skolem's Problem - On the Border , 2005 .

[70]  George D. Birkhoff,et al.  Analytic theory of singular difference equations , 1933 .

[71]  Markus Grimmer.  Interval Arithmetic in Maple with intpakX , 2003 .

[72]  T. J. Rivlin,et al.  Ultra-arithmetic I: Function data types , 1982 .

[73]  Willard L. Miranker,et al.  Validating computation in a function space , 1988 .

[74]  H. Samara,et al.  A unified approach to the Tau Method and Chebyshev series expansion techniques , 1993 .

[75]  Joris van der Hoeven,et al.  Mathemagix: the quest of modularity and efficiency for symbolic and certified numeric computation? , 2012, ACCA.

[76]  L. Fuchs,et al.  Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coefficienten. , 1866 .

[77]  Jounaïdi Abdeljaoued,et al.  Méthodes matricielles : introduction à la complexité algébrique , 2004 .

[78]  Rob Noble.  Zeros and Asymptotics of Holonomic Sequences , 2011 .

[79]  Manuel Kauers,et al.  Computer Algebra for Special Function Inequalities , 2022 .

[80]  Alexis Darrasse,et al.  Using camlp4 for presenting dynamic mathematics on the web: DynaMoW, an OCaml language extension for the run-time generation of mathematical contents and their presentation on the web , 2011, ICFP '11.

[81]  Lloyd N. Trefethen.  Computing numerically with functions instead of numbers , 2007, SNC '07.

[82]  Ramon E. Moore.  Interval arithmetic and automatic error analysis in digital computing , 1963 .

[83]  Michael Mezzino,et al.  Leibniz's Formula, Cauchy Majorants, and Linear Differential Equations , 1998 .

[84]  Robert A. van de Geijn,et al.  Anatomy of high-performance matrix multiplication , 2008, TOMS.

[85]  Joris van der Hoeven,et al.  On Effective Analytic Continuation , 2007, Math. Comput. Sci..

[86]  W. Wasow.  Asymptotic expansions for ordinary differential equations , 1965 .

[87]  Éric Schost,et al.  Power series composition and change of basis , 2008, ISSAC '08.

[88]  J. Dicapua.  Chebyshev Polynomials , 2019, Fibonacci and Lucas Numbers With Applications.

[89]  Vincent Lefèvre,et al.  MPFR: A multiple-precision binary floating-point library with correct rounding , 2007, TOMS.

[90]  Jean-Michel Muller,et al.  Handbook of Floating-Point Arithmetic (2nd Ed.) , 2018 .

[91]  Reinwald Jungen.  Sur les séries de Taylor n'ayant que des singularités algébrico-logarithmiques sur leur cercle de convergence , 1931 .

[92]  Joris van der Hoeven,et al.  Fast Evaluation of Holonomic Functions , 1999, Theor. Comput. Sci..

[93]  Marko Petkovsek,et al.  Hypergeometric Solutions of Linear Recurrences with Polynomial Coefficents , 1992, J. Symb. Comput..

[94]  Sergey V. Yakhontov.  A simple algorithm for the evaluation of the hypergeometric series using quasi-linear time and linear space , 2011, ArXiv.

[95]  G. Corliss.  Survey of interval algorithms for ordinary differential equations , 1989 .

[96]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[97]  Donald E. Knuth,et al.  Big Omicron and big Omega and big Theta , 1976, SIGA.

[98]  Arnold Schönhage,et al.  The fundamental theorem of algebra in terms of computational complexity - preliminary report , 1982 .

[99]  G. Latta,et al.  Ordinary differential equations and their solutions , 1960 .

[100]  David Barton,et al.  The Automatic Solution of Systems of Ordinary Differential Equations by the Method of Taylor Series , 1971, Computer/law journal.

[101]  Giuseppe Fiorentino,et al.  Design, analysis, and implementation of a multiprecision polynomial rootfinder , 2000, Numerical Algorithms.

[102]  R. V. Zahar.  A mathematical analysis of Miller's algorithm , 1976 .

[103]  Jet Wimp,et al.  Current trends in asymptotics: some problems and some solutions , 1991 .

[104]  Christoph Koutschan,et al.  Advanced applications of the holonomic systems approach , 2010, ACCA.

[105]  Michael B. Monagan,et al.  Maximal quotient rational reconstruction: an almost optimal algorithm for rational reconstruction , 2004, ISSAC '04.

[106]  Keith O. Geddes,et al.  Polynomial Approximation by Projections on the Unit Circle , 1975 .

[107]  J. Hoeven.  Majorants for formal power series , 2003 .

[108]  Jonathan M. Borwein,et al.  CLOSED FORMS: WHAT THEY ARE AND WHY WE CARE , 2013 .

[109]  Stefan Gerhold,et al.  Combinatorial Sequences: Non-Holonomicity and Inequalities , 2008 .

[110]  Joris van der Hoeven,et al.  Efficient accelero-summation of holonomic functions , 2007, J. Symb. Comput..

[111]  Adrien Poteaux.  Calcul de développements de Puiseux et application au calcul du groupe de monodromie d'une courbe algébrique plane , 2008 .

[112]  R. C. Whaley,et al.  Minimizing development and maintenance costs in supporting persistently optimized BLAS , 2005, Softw. Pract. Exp..

[113]  G. K. Immink.  Reduction to canonical forms and the Stokes phenomenon in theory of linear difference equations , 1991 .

[114]  Emmanuel Jeandel.  Évaluation rapide de fonctions hypergéométriques , 2000 .

[115]  A. Storjohann.  Faster algorithms for integer lattice basis reduction , 1996 .

[116]  Peter B. Borwein,et al.  On the Complexity of Calculating Factorials , 1985, J. Algorithms.

[117]  P. E. Falloon,et al.  Theory and computation of spheroidal wavefunctions , 2003 .

[118]  Richard J. Mathar.  Chebyshev series expansion of inverse polynomials , 2006 .

[119]  Arne Storjohann,et al.  The shifted number system for fast linear algebra on integer matrices , 2005, J. Complex..

[120]  Martin Campbell-Kelly,et al.  The history of mathematical tables : from Sumer to spreadsheets , 2003 .

[121]  Éric Schost,et al.  Optimization techniques for small matrix multiplication , 2011, ACCA.

[122]  Nedialko S. Nedialkov,et al.  On Taylor Model Based Integration of ODEs , 2007, SIAM J. Numer. Anal..

[123]  D. Zeilberger.  A holonomic systems approach to special functions identities , 1990 .

[124]  Moulay A. Barkatou,et al.  An Algorithm Computing the Regular Formal Solutions of a System of Linear Differential Equations , 1999, J. Symb. Comput..

[125]  L. Lipshitz,et al.  D-finite power series , 1989 .

[126]  M. Berz,et al.  TAYLOR MODELS AND OTHER VALIDATED FUNCTIONAL INCLUSION METHODS , 2003 .

[127]  Daniel Richardson,et al.  Some undecidable problems involving elementary functions of a real variable , 1969, Journal of Symbolic Logic.

[128]  Philippe Flajolet,et al.  Partial match retrieval of multidimensional data , 1986, JACM.

[129]  D. V. Chudnovsky,et al.  Approximations and complex multiplication according to Ramanujan , 2000 .

[130]  Jonathan M. Borwein,et al.  High-precision computation: Mathematical physics and dynamics , 2010, Appl. Math. Comput..

[131]  Joris van der Hoeven,et al.  Around the numeric-symbolic computation of differential Galois groups , 2007, J. Symb. Comput..

[132]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[133]  Zilin Du,et al.  Uniform complexity of approximating hypergeometric functions with absolute error , 2006 .

[134]  D. J. Bernstein.  Fast multiplication and its applications , 2008 .

[135]  Jonathan M. Borwein,et al.  Pi and the AGM , 1999 .

[136]  Jean-Pierre Serre,et al.  Über die Integration der linearen Differentialgleichungen durch Reihen , 1968 .

[137]  Jean-Michel Muller,et al.  Elementary Functions: Algorithms and Implementation , 1997 .

[138]  Manuel Bronstein,et al.  On Regular and Logarithmic Solutions of Ordinary Linear Differential Systems , 2005, CASC.

[139]  Joris van der Hoeven,et al.  Fast Evaluation of Holonomic Functions Near and in Regular Singularities , 2001, J. Symb. Comput..

[140]  Richard P. Stanley,et al.  Differentiably Finite Power Series , 1980, Eur. J. Comb..

[141]  渡辺 隼郎.  Formula manipulations solving linear ordinary differential equations , 1977 .

[142]  Manuel Kauers,et al.  A non-holonomic systems approach to special function identities , 2009, ISSAC '09.

[143]  Jrg Arndt,et al.  Matters Computational: Ideas, Algorithms, Source Code , 2010 .

[144]  L. Fox,et al.  Chebyshev polynomials in numerical analysis , 1970 .

[145]  Manuel Kauers,et al.  When can we detect that a P-finite sequence is positive? , 2010, ISSAC.

[146]  A. O. Guelfond,et al.  Calcul des Différences finies , 1964 .

[147]  D. Zeilberger,et al.  Resurrecting the asymptotics of linear recurrences , 1985 .

[148]  R. Baker Kearfott,et al.  Introduction to Interval Analysis , 2009 .

[149]  Marc Mezzarobba,et al.  NumGfun: a package for numerical and analytic computation with D-finite functions , 2010, ISSAC.

[150]  A. Odlyzko.  Asymptotic enumeration methods , 1996 .

[151]  Stanisław Lewanowicz,et al.  Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series , 1976 .

[152]  Arnold Schönhage,et al.  Fast algorithms - a multitape Turing machine implementation , 1994 .

[153]  Mark Giesbrecht,et al.  Shiftless decomposition and polynomial-time rational summation , 2003, ISSAC '03.

[154]  Martin Fürer.  Faster integer multiplication , 2007, STOC '07.

[155]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[156]  Barry Gergel,et al.  Space-efficient evaluation of hypergeometric series , 2005, SIGS.

[157]  C. Pomerance,et al.  Prime Numbers: A Computational Perspective , 2002 .

[158]  Alin Bostan,et al.  Fast algorithms for polynomial solutions of linear differential equations , 2005, ISSAC.

[159]  Keith O. Geddes.  Symbolic computation of recurrence equations for the Chebyshev series solution of linear ODE's. [ordinary differential equations , 1977 .

[160]  Doron Zeilberger,et al.  The Method of Creative Telescoping , 1991, J. Symb. Comput..

[161]  G. M.,et al.  Theory of Differential Equations , 1902, Nature.

[162]  Jean-Charles Delvenne,et al.  The continuous Skolem-Pisot problem , 2010, Theor. Comput. Sci..

[163]  Jean-Guillaume Dumas,et al.  Dense Linear Algebra over Word-Size Prime Fields: the FFLAS and FFPACK Packages , 2006, TOMS.

[164]  M. Joldes,et al.  Rigorous Polynomial Approximations and Applications , 2011 .