Fredrik Johansson
ABSTRACT
We adapt the rectangular splitting technique of Paterson and Stockmeyer to the problem of evaluating terms in holonomic sequences that depend on a parameter. This approach allows computing the n-th term in a recurrent sequence of suitable type using O(n1/2) "expensive" operations at the cost of an increased number of "cheap" operations.
Rectangular splitting has little overhead and can perform better than either naive evaluation or asymptotically faster algorithms for ranges of n encountered in applications. As an example, fast numerical evaluation of the gamma function is investigated. Our work generalizes two previous algorithms of Smith.
- A. V. Aho, K. Steiglitz, and J. D. Ullman. Evaluating polynomials at fixed sets of points. SIAM Journal on Computing, 4(4):533--539, 1975.Google Scholar
Digital Library
- D. J. Bernstein. Fast multiplication and its applications. Algorithmic Number Theory, 44:325--384, 2008.Google Scholar
- P. B. Borwein. Reduced complexity evaluation of hypergeometric functions. Journal of Approximation Theory, 50(3), July 1987. Google ScholarDigital Library
- A. Bostan, P. Gaudry, and É. Schost. Linear recurrences with polynomial coefficients and application to integer factorization and Cartier-Manin operator. SIAM Journal on Computing, 36(6):1777--1806, 2007. Google ScholarDigital Library
- R. P. Brent and P. Zimmermann. Modern Computer Arithmetic. Cambridge University Press, 2011. Google ScholarDigital Library
- D. G. Cantor and E. Kaltofen. On fast multiplication of polynomials over arbitrary algebras. Acta Informatica, 28(7):693--701, 1991. Google ScholarDigital Library
- D. V. Chudnovsky and G. V. Chudnovsky. Approximations and complex multiplication according to Ramanujan. In Ramanujan Revisited, pages 375--472. Academic Press, 1988.Google Scholar
- W. B. Hart. Fast Library for Number Theory: An Introduction. In Proceedings of the Third international congress conference on Mathematical software, ICMS'10, pages 88--91, Berlin, Heidelberg, 2010. Springer-Verlag. http://flintlib.org. Google ScholarDigital Library
- F. Johansson. Arb: a C library for ball arithmetic (ISSAC 2013 software presentation). ACM Communications in Computer Algebra, 47(4):166--169, 2013. http://fredrikj.net/arb/. Google ScholarDigital Library
- S. Köhler and M. Ziegler. On the stability of fast polynomial arithmetic. In Proceedings of the 8th Conference on Real Numbers and Computers, Santiago de Compostela, Spain, 2008.Google Scholar
- M. Mezzarobba. NumGfun: a package for numerical and analytic computation with D-finite functions. In Proceedings of ISSAC'10, pages 139--146, 2010. Google ScholarDigital Library
- M. S. Paterson and L. J. Stockmeyer. On the number of nonscalar multiplications necessary to evaluate polynomials. SIAM Journal on Computing, 2(1), March 1973.Google ScholarDigital Library
- D. M. Smith. Efficient multiple-precision evaluation of elementary functions. Mathematics of Computation, 52:131--134, 1989.Google ScholarCross Ref
- D. M. Smith. Algorithm: Fortran 90 software for floating-point multiple precision arithmetic, gamma and related functions. Transactions on Mathematical Software, 27:377--387, 2001. Google ScholarDigital Library
- J. van der Hoeven. Fast evaluation of holonomic functions. TCS, 210:199--215, 1999. Google ScholarDigital Library
- J. van der Hoeven. Making fast multiplication of polynomials numerically stable. Technical Report 2008-02, Université Paris-Sud, Orsay, France, 2008.Google Scholar
- J. von zur Gathen and J. Gerhard. Fast algorithms for Taylor shifts and certain difference equations. In Proceedings of ISSAC'97, pages 40--47, 1997. Google ScholarDigital Library
- J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, 2nd edition, 2003. Google ScholarDigital Library
- M. Ziegler. Fast (multi-)evaluation of linearly recurrent sequences: Improvements and applications. 2005. http://arxiv.org/abs/cs/0511033.Google Scholar
Index Terms
Evaluating parametric holonomic sequences using rectangular splitting
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