IMSL C Math Library
int_fcn_hyper_rect

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Integrate a function on a hyper-rectangle,
Synopsis
#include <imsl.h>
float imsl_f_int_fcn_hyper_rect (float fcn(), int ndim, float a[], float b[], …, 0)
The type double function is imsl_d_int_fcn_hyper_rect.
Required Arguments
float fcn (int ndim, float x[]) (Input)
User-supplied function to be integrated.
int ndim (Input)
The dimension of the hyper-rectangle.
float a[] (Input)
Lower limits of integration.
float b[] (Input)
Upper limits of integration.
Return Value
The value of
is returned. If no value can be computed, then NaN is returned.
Synopsis with Optional Arguments
#include <imsl.h>
float imsl_f_int_fcn_hyper_rect (float fcn(), int ndim, float a[], float b[],
IMSL_ERR_ABS, float err_abs,
IMSL_ERR_REL, float err_rel,
IMSL_ERR_EST, float *err_est,
IMSL_MAX_EVALS, int max_evals,
IMSL_FCN_W_DATA, float fcn(), void *data,
0)
Optional Arguments
IMSL_ERR_ABS, float err_abs (Input)
Absolute accuracy desired.
Default: , where ɛ is the machine precision.
IMSL_ERR_REL, float err_rel (Input)
Relative accuracy desired.
Default: , where ɛ is the machine precision.
IMSL_ERR_EST, float *err_est (Output)
Address to store an estimate of the absolute value of the error.
IMSL_MAX_EVALS, int max_evals (Input)
Number of evaluations allowed.
Default: max_evals = 32n.
IMSL_FCN_W_DATA, float fcn (int ndim, float x[], void *data), void *data (Input)
User supplied function to be integrated, which also accepts a pointer to data that is supplied by the user. data is a pointer to the data to be passed to the user-supplied function. See Passing Data to User-Supplied Functions in the introduction to this manual for more details.
Description
The function imsl_f_int_fcn_hyper_rect approximates the n-dimensional iterated integral
An estimate of the error is returned in the optional argument err_est. The approximation is achieved by iterated applications of product Gauss formulas. The integral is first estimated by a two-point tensor product formula in each direction. Then for i = 1, …, n, the function calculates a new estimate by doubling the number of points in the i-th direction, then halving the number immediately afterwards if the new estimate does not change appreciably. This process is repeated until either one complete sweep results in no increase in the number of sample points in any dimension; the number of Gauss points in one direction exceeds 256; or the number of function evaluations needed to complete a sweep exceeds max_evals.
On some platforms, imsl_f_int_fcn_hyper_rect can evaluate the user-supplied function fcn in parallel. This is done only if the function imsl_omp_options is called to flag user-defined functions as thread-safe. A function is thread-safe if there are no dependencies between calls. Such dependencies are usually the result of writing to global or static variables.
Example
In this example, we compute the integral of
on an expanding cube. The values of the error estimates are machine dependent. The exact integral over R3 is π3/2.
 
#include <imsl.h>
#include <stdio.h>
#include <math.h>
 
float fcn(int n, float x[]);
 
int main()
{
int i, j, ndim = 3;
float q, limit, a[3], b[3];
 
imsl_omp_options(
IMSL_SET_FUNCTIONS_THREAD_SAFE, 1,
0);
 
printf(" integral limit \n");
limit = pow(imsl_f_constant("pi",0), 1.5);
 
/* Evaluate the integral */
for (i = 0; i < 6; i++) {
for (j = 0; j < 3; j++) {
a[j] = -(i+1)/2.;
b[j] = (i+1)/2.;
}
q = imsl_f_int_fcn_hyper_rect (fcn, ndim, a, b,
0);
 
/* Print the result and the */
/* limiting answer */
printf(" %10.3f %10.3f\n", q, limit);
}
}
 
float fcn(int n, float x[])
{
float s;
 
s = x[0]*x[0] + x[1]*x[1] + x[2]*x[2];
return exp(-s);
}
 
Output
 
integral limit
0.785 5.568
3.332 5.568
5.021 5.568
5.491 5.568
5.561 5.568
5.568 5.568
Warning Errors
IMSL_MAX_EVALS_TOO_LARGE
The argument max_evals was set greater than 28n.
Fatal Errors
IMSL_NOT_CONVERGENT
The maximum number of function evaluations has been reached, and convergence has not been attained
IMSL_STOP_USER_FCN
Request from user supplied function to stop algorithm. User flag = "#".