Walter Gautschi

Numerical analysis: an introduction

This is a text in numerical analysis which is taken to mean the branch of mathematics that develops and analyzes computational methods dealing with problems arising in classical analysis, approximations theory, the theory of equations, and ordinary differential equations. The topics in this book are presented with a view towards stressing basic principles and maintaining simplicity and teachability as far as possible. Topics that require a level of technicality that goes beyond the standard of simplicity imposed are referenced in bibliographic notes at the end of each chapter. This book does not cover numerical linear algebra, nor the numerical solution of partial differential equations, as the author takes the view that these are now separate disciplines. It is intended that the student has a good background in calculus and advanced calculus and some knowledge of linear algebra, complex analysis, and differential equations.

Adaptive quadrature-revisited

First, the basic principles of adaptive quadrature are reviewed. Adaptive quadrature programs being recursive by nature, the choice of a good termination criterion is given particular attention. Two Matlab quadrature programs are presented. The first is an implementation of the well-known adaptive recursive Simpson rule; the second is new and is based on a four-point Gauss-Lobatto formula and two successive Kronrod extensions. Comparative test results are described and attention is drawn to serious deficiencies in the adaptive routines quad and quad8 provided by Matlab.

Numerical integration of ordinary differential equations based on trigonometric polynomials

There are many numerical methods available for the step-by-step integration of ordinary differential equations. Only few of them, however, take advantage of special properties of the solution that may be known in advance. Examples of such methods are those developed by BROCK and MURRAY [9], and by DENNIS Eg], for exponential type solutions, and a method by URABE and MISE [b~ designed for solutions in whose Taylor expansion the most significant terms are of relatively high order. The present paper is concerned with the case of periodic or oscillatory solutions where the frequency, or some suitable substitute, can be estimated in advance. Our methods will integrate exactly appropriate trigonometric polynomials of given order, just as classical methods integrate exactly algebraic polynomials of given degree. The resulting methods depend on a parameter, v=h~o, where h is the step length and ~o the frequency in question, and they reduce to classical methods if v-~0. Our results have also obvious applications to numerical quadrature. They will, however, not be considered in this paper. 1. Linear functionals of algebraic and trigonometric order In this section [a, b~ is a finite closed interval and C ~ [a, b~ (s > 0) denotes the linear space of functions x(t) having s continuous derivatives in Fa, b~. We assume C s [a, b~ normed by s (t.tt IIxll = )2 m~x Ix~ (ttt. a=0 a~t~b

On Generating Orthogonal Polynomials

We consider the problem of numerically generating the recursion coefficients of orthogonal polynomials, given an arbitrary weight distribution of either discrete, continuous, or mixed type. We discuss two classical methods, respectively due to Stieltjes and Chebyshev, and modern implementations of them, placing particular emphasis on their numerical stability properties. The latter are being studied by analyzing the numerical condition of appropriate finite-dimensional maps. A number of examples are given to illustrate the strengths and weaknesses of the various methods and to test the theory developed for them.

Remark on algorithm 726: ORTHPOL—A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules

A collection of subroutines and examples of their uses, as well as the underlying numerical methods, are described for generating orthogonal polynomials relative to arbitrary weight functions. The object of these routines is to produce the coefficients in the three-term recurrence relation satisfied by the orthogonal polynomials. Once these are known, additional data can be generated, such as zeros of orthogonal polynomials and Gauss-type quadrature rules, for which routines are also provided.

Efficient Computation of the Complex Error Function

The paper is concerned with the computation of

A Survey of Gauss-Christoffel Quadrature Formulae

w(z) = \exp ( - z^2 ){\operatorname{erfc}}( - iz)

Construction of Gauss-Christoffel quadrature formulas

for complex

An algorithm for simultaneous orthogonal transformation of several positive definite symmetric matrices to nearly diagonal form

z = x + iy

Orthogonal polynomials: applications and computation

in the first quadrant