D113 Adaptive Complex Integration Along a Line Segment

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

$I=\int$_A^Bf(z)dz.

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 (N002), ABEND (Z035), 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 $F(Z)=\; f(Z)$ .

A,B

End-points of integration interval.

EPS

Accuracy parameter (see Accuracy).

Method:

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

$\int$_a^bf(z)dz

and define

$r(a,b)\; =\{|g$₁₆(a,b) - g₈(a,b)|1+|g₁₆(a,b)|}.

Then, with G = CGAUSS or WGAUSS,

$G\; =\sum$_i=1^kg₁₆(z_i-1,z_i),

where, starting with $z$₀=A and finishing with $z$_k=B , the subdivision points $z$_i (i=1,2,...) are given by

$z$_i= z_i-1+ λ(B-z_i-1),

with $\lambda$ equal to the first member of the sequence $1,1/2,1/4,...$ for which $r(z$_i-1,z_i) < EPS . If, at any stage in the process of subdivision, the ratio

$q=|\{z$_i-z_i-1B-A}|

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

$I$_abs= ∫_A^B|f(z)|dz,

then the relation

$\{|G\; -\; I|I$_abs+k}< EPS

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.
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D114