D103: Adaptive Gaussian Quadrature

Author(s): K.S. Kölbig	Library: MATHLIB
Submitter:	Submitted: 02.05.1966
Language: Fortran	Revised: 15.03.1993

Function subprograms GAUSS, DGAUSS and QGAUSS compute, to an attempted specified accuracy, the value of the integral

The quadruple-precision version QGAUSS is available only on computers which support a REAL*16 Fortran data type.

Structure:

FUNCTION subprograms
User Entry Names: GAUSS, DGAUSS, QGAUSS
Files Referenced: Unit 6
External References: MTLMTR, ABEND, user-supplied FUNCTION subprogram

Usage:

In any arithmetic expression,

GAUSS(F,A,B,EPS), DGAUSS(F,A,B,EPS) or QGAUSS(F,A,B,EPS)

has the approximate value of the integral I.

F

Name of a user-supplied FUNCTION subprogram, declared EXTERNAL in the calling program. This subprogram must set .

A,B

End-points of integration interval. Note that B may be less than A.

EPS

Accuracy parameter (see Accuracy).

GAUSS is of type REAL, DGAUSS is of type DOUBLE PRECISION, QGAUSS is of type REAL*16, and the arguments F, A, B, EPS and X (in F) have the same type as the function name.

Method:

For any interval [a,b] we define and to be the 8-point and 16-point Gaussian quadrature approximations to

and define

Then, with G = GAUSS or DGAUSS,

where, starting with and finishing with , the subdivision points are given by

with equal to the first member of the sequence for which . If, at any stage in the process of subdivision, the ratio

is so small that 1+0.005q is indistinguishable from 1 to machine accuracy, an error exit occurs with the function value set equal to zero.

Accuracy:

Unless there is severe cancellation of positive and negative values of f(x) over the interval [A,B], the argument EPS may be considered as specifying a bound on the relative error of I in the case |I|>1, and a bound on the absolute error in the case |I|<1. More precisely, if k is the number of sub-intervals contributing to the approximation (see Method), and if

then the relation

will nearly always be true, provided the routine terminates without printing an error message. For functions f having no singularities in the closed interval [A,B] the accuracy will usually be much higher than this.

Error handling:

Error D103.1: The requested accuracy cannot be obtained (see Method). The function value is set equal to zero, and a message is written on Unit 6 unless subroutine MTLSET (N002) has been called.

Notes:

Values of the function f(x) at the interval end-points A and B are not required. The subprogram may therefore be used when these values are undefined.