Eugene V. Zima

Chains of recurrences—a method to expedite the evaluation of closed-form functions

Chains of Recurrences (CRs) are introduced as an effective method to evaluate functions at regular intervals. Algebraic properties of CRs are examined and an algorithm that constructs a CR for a given function is explained. Finally, an implementation of the method in MAXIMA/Common Lisp is discussed.

Simplification and optimization transformations of chains of recurrences

The problem of expediting the evaluation of closed-form functions at regular intervals is considered. The Chain of Recurrences technique to expedite computations is extended by rational simplifications and examined as a form of internal representation, oriented towards fast evaluation. Optimizing transformations of Chains of Recurrences are proposed.

Recurrent Relations and Speed-up of Computations Using Computer Algebra Systems

Implementation of various numerical methods often needs organisation of computation using complex iterative formulae, i.e. cycles. The time for performing a program on a computer and the necessary storage capacity are largely dependent on form of these formulae. It is desirable to construct computations so that each iterative step might use the results obtained at the previous steps as completely as possible. This implies the association of the given formulae with the recurrent relations, that bring about the same result but economise the arithmetic operations. A special algebraic method has been created that provides for the automatic construction of such recurrent relations. This method and main ways of its use in the algorithms of cycle optimisation, in the computer algebra systems and in algorithms of automatic parallel programs construction are explained. Many examples and programs are given.

Automatic construction of systems of recurrence relations

Abstract The problem of economizing on the number of arithmetic operations performed in a cycle is considered. A method is given for constructing, from the arithmetic expression giving the function computed in a cycle, systems of recurrence relations connecting the next value of the function with the results of the earlier steps of the computation.

Shiftless decomposition and polynomial-time rational summation

New algorithms are presented for computing the dispersion set of two polynomials over Q and for shiftless factorization. Together with a summability criterion by Abramov, these are applied to get a polynomial-time algorithm for indefinite rational summation, using a sparse representation of the output.

Multidimensional chains of recurrences

A technique to expedite iterative computations which is based on multidimensional chains of recurrences (MCR) is presented. Algorithms for MCR construction, interpretation and MCR-based code generation are discussed. The notion of delayed MCR simpli cation introduced here for the rst time often leads to reduced times for both the MCR construction and MCR interpretation phases of this technique. Three di erent implementations of the MCR technique (in Maple, C and Java) are described.

D'Alembertian solutions of inhomogeneous linear equations (differential, difference, and some other)

Let an Ore polynomial ring k[X; a, 6] and a nonzero pseudolinear map 19: K + K, where K is a O, &compatible extension of the field k, be given. Then we have the ring k[O] of operators K ~ K. It is assumed that if a first-order equation Fg = O, F ~ k[9], has a nonzero solution in a u, b-compatible extension of the field k, then the equation has a nonzero solution in K. These solutions form the set %,t C K of hyperexponential elements. An equation Py = O, P ~ k[O], is called completely factorable if P can be decomposed in the product of first-order operators over k. Solutions of all completely factorable equations form the linear space dk C K of d’Alembertian elements. The order of minimal operator over k which annihilates a G ~k is called the height of a. It is easy to see that %k C J& and the height of any a C %k is equal to 1. It is known ([12, 4]) that if L E ,k[@] and ~ G ‘?-l~ then all the hyperexponential solutions of the equation

On computational properties of chains of recurrences

Backward and mixed chains of recurrences are introduced. A complete set of chains of recurrences manipulation tools is described. Applications of these tools, related to the safety and numeric stability of chained computations are given.

Time-and space-efficient evaluation of some hypergeometric constants

The currently best known algorithms for the numerical evaluation of hypergeometric constants such as Ç(3) to <i>d</i> decimal digits have time complexity <i>O</i>(<i>M</i>(<i>d</i>) log²<i>d</i>) and space complexity of <i>O</i>(<i>d</i> log <i>d</i>) or <i>O</i>(<i>d</i>). Following work from Cheng, Gergel, Kim and Zima, we present a new algorithm with the same asymptotic complexity, but more efficient in practice. Our implementation of this algorithm improves over existing programs for the computation of Π, and we announce a new record of 2 billion digits for Ç(3).

Minimal completely factorable annihilators

We propose an algorithm to construct the minimal annihilating operator of a function or a sequence, when the operator is completely factorable (i.e. can be decomposed in first order factors). The algorithm is designed in the frame of the Ore rings theory and can be used in the differentiaf, difference and q-difference cases. We describe also a Maple implement ation of the algorithm.