D113: Adaptive Complex Integration Along a Line Segment

Author(s): G.A. Erskine	Library: MATHLIB
Submitter: K.S. Kölbig	Submitted: 07.12.1970
Language: Fortran	Revised: 15.03.1993

Function subprograms CGAUSS and WGAUSS compute, to an attempted specified accuracy, the value of the complex integral

The path of integration is the directed line segment AB in the complex z-plane. The function f(z) must be single-valued on this segment.

The double-precision version WGAUSS is available only on computers which support a COMPLEX*16 Fortran data type.

Structure:

FUNCTION subprograms
User Entry Names: CGAUSS, WGAUSS
Files Referenced: Unit 6
External References: MTLMTR, ABEND, user-supplied FUNCTION subprogram

Usage:

In any arithmetic expression,

CGAUSS(F,A,B,EPS) or WGAUSS(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 subroutine must set .

A,B

End-points of integration interval.

EPS

Accuracy parameter (see Accuracy).

CGAUSS is of type COMPLEX, WGAUSS is of type COMPLEX*16, and the arguments F, A, B, and Z (in F) have the same type as the function name. EPS is of type REAL for CGAUSS and of type DOUBLE PRECISION for WGAUSS.

Method:

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

and define

Then, with G = CGAUSS or WGAUSS,

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(z) 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 D113.1: The requested accuracy (see Method) cannot be obtained. 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(z) at the end-points of the line segment A and B are not required. The subprogram may therefore be used when these values are undefined.