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1. 1. Overview of Numerical Quadrature The numerical evaluation of integrals is one of the oldest problems in mathematics. One can trace its roots back at least to Archimedes. The task is to compute the value of the definite integral of a given function. This is the area under a curve in one dimension or a volume in several dimensions. In addition to being a problem of great practi­ cal interest it has also lead to the development of mathematics of much beauty and insight. Many portions of approximation theory are directly applicable to integration and results from areas as diverse as orthogo­ nal polynomials, Fourier series and number theory have had important implications for the evaluation of integrals. We denote the problem addressed here as numerical integration or numerical quadrature. Over the years analysts and engineers have contributed to a growing body of theorems, algorithms and lately, programs, for the solution of this specific problem. Much effort has been devoted to techniques for the analytic evalua­ tion of integrals. However, most routine integrals in practical scien­ tific work are incapable of being evaluated in closed form. Even if an expression can be derived for the value of an integral, often this reveals itself only after inordinate amounts of error prone algebraic manipulation. Recently some computer procedures have been developed which can perform analytic integration when it is possible.

Inhaltsverzeichnis

Frontmatter

I. Introduction

Abstract
The numerical evaluation of integrals is one of the oldest problems in mathematics. One can trace its roots back at least to Archimedes. The task is to compute the value of the definite integral of a given function. This is the area under a curve in one dimension or a volume in several dimensions. In addition to being a problem of great practical interest it has also lead to the development of mathematics of much beauty and insight. Many portions of approximation theory are directly applicable to integration and results from areas as diverse as orthogonal polynomials, Fourier series and number theory have had important implications for the evaluation of integrals. We denote the problem addressed here as numerical integration or numerical quadrature. Over the years analysts and engineers have contributed to a growing body of theorems, algorithms and lately, programs, for the solution of this specific problem.
Robert Piessens, Elise de Doncker-Kapenga, Christoph W. Überhuber, David K. Kahaner

II. Theoretical Background

Abstract
It is the objective of all automatic integrators which are part of QUADPACK to calculate a numerical approximation for the solution of a one dimensional integration problem
$$ {\text{I = }}\int\limits_a^b {{\text{bf}}({\text{x}}){\text{dx}}} $$
(2.1.1)
to within a requested absolute accuracy εa and/or a requested relative accuracy εr.
Robert Piessens, Elise de Doncker-Kapenga, Christoph W. Überhuber, David K. Kahaner

III. Algorithm Descriptions

Abstract
We shall use the term integrator to indicate either the algorithm or the integration routine. If we want to state explicitly that the double precision version of the routine is meant, we shall put a D in front of the single precision name.
Robert Piessens, Elise de Doncker-Kapenga, Christoph W. Überhuber, David K. Kahaner

IV. Guidelines for the Use of QUADPACK

Abstract
Before providing guidelines for the use of the automatic integration package in a given situation, we shall first eliminate some occasions where the application of an automatic quadrature procedure is too expensive or wasteful or even impossible.
Robert Piessens, Elise de Doncker-Kapenga, Christoph W. Überhuber, David K. Kahaner

V. Special Applications of QUADPACK

Without Abstract
Robert Piessens, Elise de Doncker-Kapenga, Christoph W. Überhuber, David K. Kahaner

VI. Implementation Notes and Routine Listings

Abstract
All the programs of QUADPACK are written in Standard FORTRAN and have been checked with the PFORT verifier of Bell Labs.
Robert Piessens, Elise de Doncker-Kapenga, Christoph W. Überhuber, David K. Kahaner

Backmatter

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