Uniform complexity of approximating hypergeometric functions with absolute error

The approximation of the general hypergeometric function H(a;b; x) = pFq(a;b; x) to any specified absolute error bound is shown to be solvable. In other words, we provide an algorithm that is uniform in the hypergeometric parameters a = (a1, . . . , ap),b = (b1, . . . , bq). An explicit bound for the complexity of our algorithm is given when the input numbers are rational. We further address the problem of evaluating H when x is a “blackbox number”, i.e., x is represented by a procedure that returns an approximation of x to any specified absolute precision. This generalization allows us to extend our approximability results to most of the familiar transcendental functions of classical analysis that are derived from H. In particular, this solves the so-called Table Maker’s Dilemma for such functions. Our algorithm has been implemented in our open-source Core Library.

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

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

[3]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[4]  Kurt Mehlhorn,et al.  Certifying and repairing solutions to large LPs how good are LP-solvers? , 2003, SODA '03.

[5]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[6]  Arnold Schönhage,et al.  Storage Modification Machines , 1979, SIAM J. Comput..

[7]  Stephen Smale,et al.  Some Remarks on the Foundations of Numerical Analysis , 1990, SIAM Rev..

[8]  Zilin Du,et al.  Guaranteed Precision for Transcendental and Algebraic Computation Made Easy , 2006 .

[9]  Chee-Keng Yap,et al.  Randomized Zero Testing of Radical Expressions and Elementary Geometry Theorem Proving , 2000, Automated Deduction in Geometry.

[10]  Chee-Keng Yap,et al.  Theory of Real Computation According to EGC , 2008, Reliable Implementation of Real Number Algorithms.

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

[12]  Christian P. Ullrich,et al.  Computer Arithmetic and Self-Validating Numerical Methods , 1990, Notes and reports in mathematics in science and engineering.

[13]  Mark Nardin,et al.  Algorithm 707: CONHYP: a numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes , 1992, TOMS.

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

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

[16]  Kurt Mehlhorn,et al.  A Separation Bound for Real Algebraic Expressions , 2001, Algorithmica.

[17]  Walter Gautschi,et al.  NUMERICAL EVALUATION OF SPECIAL FUNCTIONS , 2001 .

[18]  Chee Yap.  On guaranteed accuracy computation , 2004 .

[19]  Peter Wittwer,et al.  Computer-Assisted Proofs in Analysis and Programming in Logic: A case Study , 1996, SIAM Rev..

[20]  Deok-Soo Kim,et al.  Shortest Paths for Disc Obstacles , 2004, ICCSA.

[21]  Ker-I Ko,et al.  Complexity Theory of Real Functions , 1991, Progress in Theoretical Computer Science.

[22]  G. M. Clemence,et al.  Blow up of smooth solutions to the barotropic compressible magnetohydrodynamic equations with finite mass and energy , 2008, 0811.4359.

[23]  Mark Nardin,et al.  Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes , 1992 .

[24]  Arnaud Tisserand,et al.  Towards correctly rounded transcendentals , 1997, Proceedings 13th IEEE Sympsoium on Computer Arithmetic.

[25]  Chee-Keng Yap,et al.  A core library for robust numeric and geometric computation , 1999, SCG '99.

[26]  Bruce D. McCullough,et al.  Assessing the Reliability of Statistical Software: Part I , 1998 .

[27]  José E. Moreira,et al.  Hypergeometric Functions in Exact Geometric Computation , 2002, CCA.

[28]  Chee-Keng Yap,et al.  Approximate Euclidean Shortest Paths in 3-Space , 1997, Int. J. Comput. Geom. Appl..

[29]  Yudell L. Luke,et al.  Algorithms for the Computation of Mathematical Functions , 1977 .

[30]  Klaus Weihrauch,et al.  Computable Analysis , 2005, CiE.