IDL Analyst Reference Guide: Quadrature

IMSL_INTFCN:
Two-dimensional Iterated Integrals

This version of the IMSL_INTFCN function integrates two-dimensional iterated integrals.

Syntax

Result = IMSL_INTFCN(f, a, b, g, h, /TWO_DIMENSIONAL)

Return Value

The value of:

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.

g

Scalar string specifying the name of a user-supplied IDL Analyst function used to evaluate the lower limit of the inner integral. Function g accepts one scalar parameter and returns a single scalar of the same type.

h

Scalar string specifying the name of a user-supplied IDL Analyst function used to evaluate the upper limit of the inner integral. Function h accepts one scalar parameter and returns a single scalar of the same type.

Keywords

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

TWO_DIMENSIONAL

Set this keyword to integrate a two-dimensional iterated integral.

Discussion

This method approximates the following two-dimensional iterated integral:

The lower-numbered rules are used for less smooth integrands, while the higher-order rules are more efficient for smooth (oscillatory) integrands.

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

Example

This example computes the value of the integral:

.RUN  
; Define the function to be integrated.  
FUNCTION f, x, y  
   RETURN, SIN(x + y)  
END  
  
.RUN  
; Define the function for the lower limit of the inner integral.  
FUNCTION g, x  
   RETURN, x  
END  
  
.RUN  
; Define the function for the upper limit of the inner integral.  
FUNCTION h, x  
   RETURN, 2 * x  
END  
  
ans = IMSL_INTFCN('f',0,1,'g','h',/Two_Dimensional)  
; Call IMSL_INTFCN with keyword Two_Dimensional set and the names  
; of the functions defining the limits of the inner integral.  
PM, 'Computed Answer:', ans  
Computed Answer:  
        0.407609  
exact = -SIN(3)/3 + SIN(2)/2  
   PM, 'Exact - Computed:', exact - ans  
Exact - Computed:  
    -5.96046e-08