D104 Cauchy Principal Value Integration

Function subprograms RCAUCH and DCAUCH compute the Cauchy principal value integral

$I\; =\; P\; \int$_A^Bf(x)dx

where f has a singularity inside or outside the interval $[A,B]$ such that the Cauchy principal value exists.

On computers other than CDC or Cray, only the double-precision version DCAUCH is available. On CDC and Cray computers, only the single-precision version RCAUCH is available.

Structure:

FUNCTION subprograms
User Entry Names: RCAUCH, DCAUCH
Obsolete User Entry Names: CAUCHY $\equiv$ RCAUCH
Files Referencend: Unit 6
External References: GAUSS, DGAUSS (D103), MTLMTR (N002), ABEND (Z035),
user-supplied FUNCTION subprogram

Usage:

For $t=R$ (type REAL), $t=D$ (type DOUBLE PRECISION),

    tCAUCH(F,A,B,S,EPS)

F

(type according to t) Name of a user-supplied FUNCTION subprogram, declared EXTERNAL in the calling program. This subprogram must set $F(X)=\; f(X)$ .

A,B

(type according to t) End-points of the integration interval. Note that B may be less than A.

S

(type according to t) The absissa of the singularity.

EPS

(type according to t) Accuracy parameter (see under Accuracy in the in short write-up for GAUSS (D103)).

Method:

The method described in Ref. 1 is used for decomposition of the integral. The resulting integrals are computed by GAUSS (D103).

Accuracy:

See short write-up for GAUSS (D103).

Error handling:

Error D104.1: $S=A$ or $S=B$ .
Error D104.2: The requested accuracy cannot be obtained (see short write-up for GAUSS (D103)).
The function value is set equal to zero, and a message is written on Unit 6, unless subroutine MTLSET (N002) has been called.

References:

1. I.M. Longman, On the numerical evaluation of Cauchy principal values of integrals, MTAC (later renamed Math. Comp.) 12 (1958) 205--207.

D105