Solution Of Linear Recurrence Relations Research Articles

Plancherel-Rotach type asymptotics for solutions of linear recurrence relations with rational coefficients

The asymptotic behaviour of solutions of difference equations with respect to the variable n with spectral parameter x is investigated. A new method for finding asymptotic expansions for basis solutions in overlapping domains of the (n,x)-space which extend to infinity is proposed. In principle, matching the expansions in the intersection of these domains makes it possible to determine the global asymptotic picture of the behaviour of solutions of equations in the complex plane of the spectral parameter x for suitable scaling depending on n. The potential of the method is demonstrated using the examples of the Hermite and Meixner polynomials. Bibliography: 27 titles.

Computing Hypergeometric Solutions of Linear Recurrence Equations

We describe a complete algorithm to compute the hypergeometric solutions of linear recurrence relations with rational function coefficients. We use the notion of finite singularities and avoid computations in splitting fields. An implementation is available in Maple 9.

A Lower Bound for Orthogonal Polynomials with an Application to Polynomial Hypergroups

We give a lower bound for solutions of linear recurrence relations of the form zan = ∑n+Nk=n−N αk, nak, whenever z is not in the lp-spectrum of the corresponding banded operator. In particular if Pn, are polynomials orthonormal with respect to a measure μ supported in a bounded interval the sequence Pn(x)2 + Pn+1(x)2 is bounded from below by (1 + ϵ)n, for x ∉ supp μ. We give an application to polynomial hypergroups.

Orthogonal polynomials, associated polynomials and functions of the second kind

A survey is given of the interaction between orthogonal polynomials, associated polynomials and functions of the second kind with an emphasis on asymptotic results. Various formulas are presented in a unified way in terms of Wronskians of solutions of linear recurrence relations. Some of these formulas are classical and go back to the previous century, but usually they are hard to locate in the literature. Some new formulas are also given, in particular a formula expressing the derivative of an orthogonal polynomial in terms of the orthogonal polynomials and the associated polynomials.

Using qr-decompositions in the numerical solution of linear recurrence relations

For the numerically stable calculation of a solution of a linear recurrence relation the degree of minimality of this solution has to be known. One method to determine the degree of minimality is by calculating the asymptotic behavior of all solutions, a task which is difficult to accomplish for many types of recurrence relations. In this paper we describe a method which uses QR-decomposition for the determination of the degree of minimality

A note on the numerical solution of linear recurrence relations

A numerical algorithm for the computation of non-dominant solutions of linear recurrence relations is analysed. Several non-trivial improvements are made and the efficiency of the new algorithm is illustrated by means of some numerical examples.

An Extension of Olver's Method for the Numerical Solution of Linear Recurrence Relations

An Extension of Olver's Method for the Numerical Solution of Linear Recurrence Relations

An extension of Olver’s method for the numerical solution of linear recurrence relations

An algorithm is developed for computing the solution of a class of linear recurrence relations of order greater than two when unstable error propagation prevents the required solution being found by direct forward recurrence. By abandoning an appropriate number of initial conditions the original problem may be replaced by an inexact but well-conditioned boundary value problem, and in certain circumstances the solution of this new problem is a good approximation to the required solution of the original problem. The required solution of this reposed problem is generated using an algorithm based on Gaussian elimination, and a technique developed by Olver is extended to estimate automatically the truncation error of the proposed algorithm.

An Efficient Parallel Algorithm for the Solution of a Tridiagonal Linear System of Equations

Tridiagonal linear systems of equations can be solved on conventional serial machines in a time proportional to N , where N is the number of equations. The conventional algorithms do not lend themselves directly to parallel computation on computers of the ILLIAC IV class, in the sense that they appear to be inherently serial. An efficient parallel algorithm is presented in which computation time grows as log 2 N . The algorithm is based on recursive doubling solutions of linear recurrence relations, and can be used to solve recurrence relations of all orders.

The numerical solution of linear recurrence relations

By analysing the effect of rounding error on linear recurrence relations having exponential-type complementary solutions, the distribution of two-point boundary conditions most likely to specify a well-conditioned problem is deduced. An intuitive explanation is derived by extending the Bernouilli method of polynomial factorisation to the case of variable coefficients. The replacement of specific boundary conditions by a knowledge of the relative behaviour of the particular and complementary solutions is discussed, and illustrated by examples.

Relative error propagation in the recursive solution of linear recurrence relations

This paper is concerned with the numerical solution of the general initial value problem for linear recurrence relations. An error analysis of direct recursion is given, based on relative rather than absolute error, and a theory of relative stability developed.Miller's algorithm for second order homogeneous relations is extended to more general cases, and the propagation of errors analysed in a similar manner. The practical significance of the theoretical results is indicated by applying them to particular classes of problem.