int_fcn_fourier
Computes a Fourier sine or cosine transform.
Synopsis
#include <imsl.h>
float imsl_f_int_fcn_fourier (float fcn(), float a, Imsl_quad weight, float omega, …, 0)
The type double function is imsl_d_int_fcn_fourier.
Required Arguments
float fcn (float x) (Input)
User-supplied function to be integrated.
float a (Input)
Lower limit of integration. The upper limit of integration is ∞.
Imsl_quad weight and float omega (Input)
These two parameters are used to describe the trigonometric weight. The parameter weight can take on the two values described below, and the parameter omega = ω specifies the frequency of the trigonometric weighting function.
|
weight |
Integration Weight |
|
IMSL_COS |
cos (ωx) |
|
IMSL_SIN |
sin (ωx) |
Return Value
The return value is
if weight = IMSL_COS. If weight = IMSL_SIN, then the cosine factor is replaced with a sine factor. If no value can be computed, NaN is returned.
Synopsis with Optional Arguments
#include <imsl.h>
float imsl_f_int_fcn_fourier (float fcn(), float a, Imsl_quad weight, float omega,
IMSL_ERR_ABS, float err_abs,
IMSL_ERR_EST, float *err_est,
IMSL_MAX_SUBINTER, int max_subinter,
IMSL_MAX_CYCLES, int max_cycles,
IMSL_MAX_MOMENTS, int max_moments,
IMSL_N_CYCLES, int *n_cycles,
IMSL_N_EVALS, int *n_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_EST, float *err_est (Output)
Address to store an estimate of the absolute value of the error.
IMSL_MAX_SUBINTER, int max_subinter (Input)
Number of subintervals allowed.
Default: max_subinter = 500
IMSL_MAX_CYCLES, int max_cycles (Input)
Number of cycles allowed.
Default: max_subinter = 50
IMSL_MAX_MOMENTS, int max_moments (Input)
Number of subintervals allowed in the partition of each cycle.
Default: max_moments = 21
IMSL_N_CYCLES, int *n_cycles (Output)
Address to store the number of cycles generated.
IMSL_N_EVALS, int *n_evals (Output)
Address to store the number of evaluations of fcn.
IMSL_FCN_W_DATA, float fcn (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 the Introduction, Passing Data to User-Supplied Functions at the beginning of this manual for more details.
Description
The function imsl_f_int_fcn_fourier 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 cosωx or sinωx. The integration interval is always semi-infinite of the form [a, ∞]. These Fourier integrals are approximated by repeated calls to the function imsl_f_int_fcn_trig followed by extrapolation.
On some platforms,imsl_f_int_fcn_fourier 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.
The function imsl_f_int_fcn_fourier is based on the subroutine QAWF by Piessens et al. (1983).
Examples
Example 1
The value of
is computed. Notice that the integrand is coded to protect for the singularity at zero.
#include <imsl.h>
#include <stdio.h>
#include <math.h>
float fcn(float x);
int main()
{
float q, exact, omega;
omega = imsl_f_constant("pi",0) / 2.;
imsl_omp_options(
IMSL_SET_FUNCTIONS_THREAD_SAFE, 1,
0);
/* Evaluate the integral */
q = imsl_f_int_fcn_fourier (fcn, 0.0,
IMSL_COS, omega,
0);
/* Print the result and the */
/* exact answer */
exact = 1.0;
printf("integral = %10.3f\nexact = %10.3f\n", q, exact);
}
float fcn(float x)
{
return (x==0.) ? 0. : 1./sqrt(x);
}
Output
integral = 1.000
exact = 1.000
The value of
is again computed. The values of the actual and estimated error are printed as well. Note that these numbers are machine dependent. Furthermore, the error estimate is usually pessimistic. That is, the actual error is usually smaller than the error estimate,
as is the case in this example.The number of function evaluations also are printed. Notice that the integrand is coded to protect for the singularity at zero.
#include <imsl.h>
#include <stdio.h>
#include <math.h>
float fcn(float x);
int main()
{
int n_evals;
float q, exact, omega, err_est, exact_err;
omega = imsl_f_constant("pi",0) / 2.0;
imsl_omp_options(
IMSL_SET_FUNCTIONS_THREAD_SAFE, 1,
0);
/* Evaluate the integral */
q = imsl_f_int_fcn_fourier (fcn, 0.0,
IMSL_COS, omega,
IMSL_ERR_EST, &err_est,
IMSL_N_EVALS, &n_evals,
0);
/* Print the result and the */
/* exact answer */
exact = 1.;
exact_err = fabs(exact - q);
printf("integral = %10.3f\nexact = %10.3f\n", q, exact);
printf("error estimate = %e\nexact error = %e\n", err_est,
exact_err);
printf("The number of function evaluations = %d\n", n_evals);
}
float fcn(float x)
{
return (x==0.) ? 0. : 1./sqrt(x);
}
Output
integral = 1.000
exact = 1.000
error estimate = 1.803637e-04
exact error = 1.013279e-06
The number of function evaluations = 405
Warning Errors
|
IMSL_BAD_INTEGRAND_BEHAVIOR |
Bad integrand behavior occurred in one or more cycles. |
|
IMSL_EXTRAPOLATION_PROBLEMS |
Extrapolation table constructed for convergence acceleration of the series formed by the integral contributions of the cycles does not converge to the requested accuracy. |
Fatal Errors
|
IMSL_MAX_CYCLES |
Maximum number of cycles allowed has been reached. |
|
IMSL_STOP_USER_FCN |
Request from user supplied function to stop algorithm. User flag = "#". |
