Joris van der Hoeven

Mathematical Software - ICMS 2006

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Fast evaluation of holonomic functions

Abstract A holonomic function is an analytic function, which satisfies a linear differential equation with polynomial coefficients. In particular, the elementary functions exp, log, sin, etc. and many special functions like erf, Si, Bessel functions, etc. are holonomic functions. Given a holonomic function f (determined by the linear differential equation it satisfies and initial conditions in a non singular point z ), we show how to perform arbitrary precision evaluations of f at a non singular point z ′ on the Riemann surface of f , while estimating the error. Moreover, if the coefficients of the polynomials in the equation for f are algebraic numbers, then our algorithm is asymptotically very fast: if M ( n ) is the time needed to multiply two n digit numbers, then we need a time O ( M ( n log 2 n log log n )) to compute n digits of f ( z ′).

Fast Evaluation of Holonomic Functions Near and in Regular Singularities

A holonomic function is an analytic function, which satisfies a linear differential equationLf= 0 with polynomial coefficients. In particular, the elementary functions exp,log,sin, etc., and many special functions such as erf, Si, Bessel functions, etc., are holonomic functions. In a previous paper, we have given an asymptotically fast algorithm to evaluate a holonomic function f at a non-singular point z? on the Riemann surface of f, up to any number of decimal digits while estimating the error. However, this algorithm becomes inefficient, when z? approaches a singularity of f. In this paper, we obtain efficient algorithms for the evaluation of holonomic functions near and in singular points where the differential operator L is regular (or, slightly more generally, where L is quasi-regular?a concept to be introduced below).

Newton's method and FFT trading

Let C[[z]] be the ring of power series over an effective ring C. In Brent and Kung (1978), it was first shown that differential equations over C[[z]] may be solved in an asymptotically efficient way using Newtons method. More precisely, if M(n) denotes the complexity for multiplying two polynomials of degree

Lazy multiplication of formal power series

For most fast algorithms to manipulate formal power series, a fast multiplication algorithm is essential. If one desires to compute all coe cients of a product of two power series up to a given order, then several e cient algorithms are available, such as fast Fourier multiplication. However, one often needs a lazy multiplication algorithm, for instance when the product computation is part of the computation of the coe cients of an implicitly de ned power series. In this paper, we describe two lazy multiplication algorithms, which are faster than the naive method. In particular, we give an algorithm of time complexity O(n log 2 n).

Faster Polynomial Multiplication over Finite Fields

Polynomials over finite fields play a central role in algorithms for cryptography, error correcting codes, and computer algebra. The complexity of multiplying such polynomials is still a major open problem. Let p be a prime, and let Mp(n) denote the bit complexity of multiplying two polynomials in Fp[X] of degree less than n. For n large compared to p, we establish the bound Mp(n) = O(n log n 8log* n log p), where log* n = min{k ϵ N: log …k×… log n ≤ 1} stands for the iterated logarithm. This improves on the previously best known bound Mp(n) = O(n log n log log n log p), which essentially goes back to the 1970s.

On Effective Analytic Continuation

Abstract.Until now, the area of symbolic computation has mainly focused on the manipulation of algebraic expressions. It would be interesting to apply a similar spirit of “exact computations” to the field of mathematical analysis.One important step for such a project is the ability to compute with computable complex numbers and computable analytic functions. Such computations include effective analytic continuation, the exploration of Riemann surfaces and the study of singularities. This paper aims at providing some first contributions in this direction, both from a theoretical point of view (such as precise definitions of computable Riemann surfaces and computable analytic functions) and a practical one (how to compute bounds and analytic continuations in a reasonably efficient way).We started to implement some of the algorithms in the MMXLIB library. However, during the implementation, it became apparent that further study was necessary, giving rise to the present paper.

FFT-like Multiplication of Linear Differential Operators

It is well known that integers or polynomials can be multiplied in an asymptotically fast way using the discrete Fourier transform. In this paper, we give an analogue of fast Fourier multiplication in the ring of skew polynomials Cx, ? , where ?=x??x. More precisely, we show that the multiplication problem of linear differential operators of degree n in x and degree n in ? can be reduced to the n×n matrix multiplication problem.

On the bit-complexity of sparse polynomial and series multiplication

In this paper we present various algorithms for multiplying multivariate polynomials and series. All algorithms have been implemented in the C++ libraries of the Mathemagix system. We describe naive and softly optimal variants for various types of coefficients and supports and compare their relative performances. For the first time, under the assumption that a tight superset of the support of the product is known, we are able to observe the benefit of asymptotically fast arithmetic for sparse multivariate polynomials and power series, which might lead to speed-ups in several areas of symbolic and numeric computation. For the sparse representation, we present new softly linear algorithms for the product whenever the destination support is known, together with a detailed bit-complexity analysis for the usual coefficient types. As an application, we are able to count the number of the absolutely irreducible factors of a multivariate polynomial with a cost that is essentially quadratic in the number of the integral points in the convex hull of the support of the given polynomial. We report on examples that were previously out of reach.

Asymptotic expansions of exp-log functions

We give an algorithm to compute asymptotic expansions of exp-log functions. This algorithm automatically computes the necessary asymptotic scale and does not suffer from problems of indefinite cancellation. In particular, an asymptotic equivalent can always be computed for a given exp-log function.