IMSL_INTFCN:
Integrals in the Cauchy Principle Value Sense

Syntax | Return Value | Arguments | Keywords | Discussion | Example | Errors

This version of the IMSL_INTFCN function computes integrals of the form:

IMSL_INTFCN_Integrals_in_the_Cauchy_Principle_Value_Sense-31.jpg

in the Cauchy principal value sense.

Note
The Singular_Pt argument and the CAUCHY keyword must be supplied to use this integration method.

Syntax

Result = IMSL_INTFCN(f, a, b, Singular_Pt, /CAUCHY)

Return Value

The value of:

IMSL_INTFCN_Integrals_in_the_Cauchy_Principle_Value_Sense-32.jpg

is returned. If no value can be computed, the floating-point value NaN (Not a Number) is returned.

Arguments

f

A scalar string specifying the name of a user-supplied function to be integrated. The function f accepts one scalar parameter and returns a single scalar of the same type.

a

A scalar expression specifying the lower limit of integration.

b

A scalar expression specifying the upper limit of integration.

Singular_Pt

A scalar expression specifying the singular point. The singular point must not equal a or b.

Keywords

In addition to the global IMSL_INTFCN keywords listed in the main section under Keywords, the following keywords may be specified:

CAUCHY

Set this keyword to compute the specified integral in the Cauchy principal value sense.

Discussion

This method uses a globally adaptive scheme in an attempt to reduce the absolute error. It computes integrals whose integrands have the special form w (x) f (x), where w (x) = 1/(x – Singular_Pt). If Singular_Pt lies in the interval of integration, then the integral is interpreted as a Cauchy principal value. A combination of modified Clenshaw-Curtis and Gauss-Kronrod formulas is employed. The method is an implementation of the subroutine QAWC by Piessens et al. (1983).

If this method is used, the function should be coded to protect endpoint singularities if they exist.

Example

The Cauchy principal value of:

IMSL_INTFCN_Integrals_in_the_Cauchy_Principle_Value_Sense-33.jpg

is computed. 

.RUN 
; Define the function to be integrated. 
FUNCTION f, x 
   RETURN, 1/(5 * x^3 + 6) 
END 
 
ans = IMSL_INTFCN('f', -1, 5, 0, /Cauchy) 
; Call IMSL_INTFCN with keyword Cauchy set. 
PM, 'Computed Answer:', ans 
; Output the results. 
Computed Answer: 
      -0.0899440 
exact = ALOG(125/631.)/18 
   PM, 'Exact - Computed:', exact - ans 
Exact - Computed: 
       1.49012e-08

Errors

See Errors.