Peter R. Graves-Morris

An extension of a row convergence theorem for vector Pade´ approximants

Abstract A row convergence theorem of de Montessus type is established for vector Pade approximants to a vector-valued meromorphic function f. While in a previous paper the authors established such a theorem for the case when f has simple poles, the essential feature of the present paper is to treat the situation when f has multiple poles.

A “Look-around Lanczos” algorithm for solving a system of linear equations

Two algorithms for the solution of a large sparse linear system of equations are proposed. The first is a modification of Lanczos method and the second is based on one of Brezinskis methods. Both the latter methods are iterative and they can break down. In practical situations, serious numerical error is far more likely to occur because an ill-conditioned pair of polynomials is (implicitly) used in the calculation rather than complete breakdown arising because a large square block of exactly defective polynomials is encountered. The algorithms proposed use a method based on selecting well-conditioned pairs of neighbouring polynomials (in the associated Padé table), and the method is equivalent to going round the blocks instead of going across them, as is done in the well-known look-ahead methods.

The rise and fall of the vector epsilon algorithm

The performance of the vector epsilon algorithm is governed by two important mathematical theorems which are briefly reviewed in context. We note that the performance of the vector epsilon algorithm is inevitably qualitatively incorrect for sequences whose generating functions have poles near unity. This difficulty is avoided by the use of hybrid vector Padé approximants.

Avoiding breakdown in Van der Vorst's method

Van der Vorsts method is a development of Lanczos iterative method for the solution of a large sparse system of linear equations. Both methods can suffer from Lanczos breakdown. The usual cure for this problem is a look-ahead method. Recently, the look-around method has been proposed, which tracks the edges of blocks in degenerate cases instead of jumping across them. Here we show how Van der Vorsts minimal residual principle can be built into the look-around method.

Solution of integral equations using function-valued Padé approximants II

AbstractWe consider the use of functional (i.e. function-valued) Padé approximants to accelerate the convergence of Neumann series of linear integral equations and to estimate their characteristic values and eigenfunctions.We apply our methods to the Neumann series solution for the linear integral equationn

A new approach to acceleration of convergence of a sequence of vectors

phi (x) = 1 + lambda int_{ - 1}^1 {K(x,y)phi (y) dy,}

Reliability of Lanczos-Type Product Methods from Perturbation Theory

n wheren