IMSL_INTFCN:
Functions Containing a Sine or Cosine Factor

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

This version of the IMSL_INTFCN function integrates functions containing a sine or a cosine factor.

Note
The Omega argument and one of the SINE, or COSINE keywords must be supplied to use this integration method.

Syntax

Result = IMSL_INTFCN(f, a, b, Omega
, /SINE | /COSINE [, MAX_MOMENTS=value])

Return Value

The value of:

IMSL_INTFCN_Functions_Containing_a_Sine_or_Cosine_Factor-26.jpg

where the weight function w (ωx) is defined by the keywords below, 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.

Omega

A scalar expression specifying the frequency of the trigonometric weighting function.

Keywords

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

SINE

Set this keyword to use sin (ωx) for the integration weight function. If SINE is supplied, COSINE must not be present.

COSINE

Set this keyword to use cos (ωx) for the integration weight function. IF COSINE is supplied, SINE must not be present.

MAX_MOMENTS

Set this keyword equal to a scalar expression specifying an upper bound on the number of Chebyshev moments that can be stored. Increasing (decreasing) this number may increase (decrease) execution speed and space used. Default: MAX_MOMENTS = 21

Discussion

This method is a special-purpose integrator that uses a globally adaptive scheme to reduce the absolute error. It computes integrals whose integrands have the special form w (x) f (x), where w (x) is either cos (ωx) or sin (ωx). Depending on the length of the subinterval in relation to the size of ω, either a modified Clenshaw-Curtis procedure or a Gauss-Kronrod 7/15 rule is employed to approximate the integral on a subinterval. This method is based on the subroutine QAWO by Piessens et al. (1983).

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

Example

The value of:

IMSL_INTFCN_Functions_Containing_a_Sine_or_Cosine_Factor-27.jpg

is computed. The exact answer is:

IMSL_INTFCN_Functions_Containing_a_Sine_or_Cosine_Factor-28.jpg

.RUN 
; Define the function to be integrated. 
FUNCTION f, x 
   RETURN, x^2 
END 
 
ans = IMSL_INTFCN('f', 0, 1, 3 * !Pi, /Sine) 
; Call IMSL_INTFCN with Sine set and value for method  
; parameter omega. 
PM, 'Computed Answer:', ans 
; Output the results. 
Computed Answer: 
       0.101325 
exact = ((3 * !Pi)^2 - 2)/((3 * !pi)^3) - 2/(3 * !Pi)^3 
PM, 'Exact - Computed:', exact - ans 
Exact - Computed: 
       0.00000

Errors

See Errors.