Avram Sidi

Extrapolation methods for vector sequences

This is an expository paper that describes and compares five methods for extrapolating to the limit (or anti-limit) of a vector sequence without explicit knowledge of the sequence generator. The methods are the minimal polynomial extrapolation (MPE) studied by Cabay and Jackson, Mesina, and Skelboe; the reduced rank extrapolation (RRE) of Eddy (which we show to be equivalent to Mesina’s version of MPE); the vector and scalar versions of the epsilon algorithm (VEA, SEA) introduced by Wynn and extended by Brezinski and Gekeler; and the topological epsilon algorithm (TEA) of Brezinski. We cover the derivation and error analysis of iterated versions of the algorithms, as applied to both linear and nonlinear problems, and we show why these versions tend to converge quadratically. We also present samples from extensive numerical testing that has led us to the following conclusions: (a) TEA, in spite of its role as a theoretical link between the polynomial-type and the epsilon-type methods, has no practical appl...

Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.

A New Variable Transformation for Numerical Integration

Presently, variable transformations are used to enhance the performance of lattice rules for multidimensional integration. The transformations that are in the literature so far are of either polynomial or exponential nature. Following a short survey of some of the transformations that have been found to be effective, we propose a new transformation, denoted the sin m -transformation, that is neither polynomial nor exponential, but trigonometric, in nature. This transformation is also a representative of a general class of variable transformations that we denote S m . We analyze the effect of transformations in S m within the framework of one-dimensional integration, and show that they have some very interesting and useful properties. Present results indicate that transformations in S m can be more advantageous than known polynomial transformations, and have less underflow and overflow problems than exponential ones. Indeed, the various numerical tests performed with the sin m -transformation support this. We end the paper with numerical examples through which some of the theory is verified.

Acceleration of convergence of vector sequences

A general approach to the construction of convergence acceleration methods for vector sequences is proposed. Using this approach, one can generate some known methods, such as the minimal polynomial extrapolation, the reduced rank extrapolation, and the topological epsilon algorithm, and also some new ones. Some of the new methods are easier to implement than the known methods and are observed to have similar numerical properties. The convergence analysis of these new methods is carried out, and it is shown that they are especially suitable for accelerating the convergence of vector sequences that are obtained when one solves linear systems of equations iteratively. A stability analysis is also given, and numerical examples are provided. The convergence and stability properties of the topological epsilon algorithm are likewise given.

An Algorithm for a Generalization of the Richardson Extrapolation Process

In this paper we present a recursive method, designated the

The numerical evaluation of very oscillatory infinite integrals by extrapolation

W^{(m)}

Efficient implementation of minimal polynomial and reduced rank extrapolation methods

algorithm, for implementing a generalization of the Richardson extrapolation process that has been ntroduced in [8]. Compared to the direct solution of the linear systems of equations defining the extrapolation procedure, this method requires a small number of arithmetic operations and very little storage. The technique is also applied to solve recursively the coefficient problem associated with the rational approximations obtained by applying the d-transformation of [6], [13] to power series. In the course of development a new recursive algorithm for implementing a very general extrapolation procedure is introduced, which is similar to that given in [2], [4] for solving the same problem. A FORTRAN program for the

An algorithm for a special case of a generalization of the Richardson extrapolation process

W^{(m)}

Convergence and stability properties of minimal polynomial and reduced rank extrapolation algorithms

algorithm is also appended.

Convergence properties of some nonlinear sequence transformations

Recently the author has given two modifications of a nonlinear extrapolation method due to Levin and Sidi, which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity. In this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity. The new method is applied to a number of complicated integrals, among them the solution to a problem in viscoelasticity. Some convergence results for this method are presented.