.p>.CMCH4.DOC!INT_FCN_TRIG;int_fcn_trig
Integrates a function containing a sine or a cosine factor.
Synopsis
#include <imsl.h>
float imsl_f_int_fcn_trig (float fcn(), float a, float b, Imsl_quad weight, float omega, ¼, 0)
The type double function is imsl_d_int_fcn_trig.
Required Arguments
float fcn (float
x)
(Input)
User-supplied function to be integrated.
float a
(Input)
Lower limit of integration.
float b
(Input)
Upper limit of integration.
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 (wx) |
|
IMSL_SIN |
sin (wx) |
Return Value
The value of
is returned 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_trig (float fcn(), float a, float b,
Imsl_quad weight, float omega,
IMSL_ERR_ABS, float
err_abs,
IMSL_ERR_REL, float err_rel,
IMSL_ERR_EST, float
*err_est,
IMSL_MAX_SUBINTER, int max_subinter,
IMSL_N_SUBINTER,
int *n_subinter,
IMSL_N_EVALS, int *n_evals,
IMSL_MAX_MOMENTS, int max_moments,
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_SUBINTER, int
max_subinter (Input)
Number of subintervals
allowed.
Default:
max_subinter = 500
IMSL_N_SUBINTER, int
*n_subinter (Output)
Address to store the number of
subintervals generated.
IMSL_N_EVALS, int *n_evals
(Output)
Address to store the number of evaluations of fcn.
IMSL_MAX_MOMENTS, int
max_moments (Input)
This is 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
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_trig 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 function uses the general strategy of the function imsl_f_int_fcn_sing. The function imsl_f_int_fcn_trig is based on the subroutine QAWO by Piessens et al. (1983).
Examples
Example 1
The value of
is computed. Notice that we have coded around the singularity at zero. This is necessary since this procedure evaluates the integrand at the two endpoints.
#include <math.h>
#include
<imsl.h>
float
fcn(float x);
main()
{
float q, exact,
omega;
omega = 10*imsl_f_constant("pi",
0);
/* Evaluate the integral */
q = imsl_f_int_fcn_trig (fcn,
0.0,
1.0,
IMSL_SIN,
omega,
0);
/* Print the result and the
*/
/* exact answer */
exact =
-.1281316;
printf("integral =
%10.3f\nexact = %10.3f\n", q, exact);
}
float
fcn(float x)
{
return (x==0.0) ? 0.0
: log(x);
}
Output
integral =
-0.128
exact = -0.128
Example 2
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, it is usually the case that the error estimate is 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 are also printed.
#include <math.h>
#include
<imsl.h>
float
fcn(float x);
main()
{
int
n_evals;
float q,
exact, omega, err_est, exact_err;
omega =
10*imsl_f_constant("pi",
0);
/* Evaluate the integral */
q = imsl_f_int_fcn_trig (fcn,
0.0,
1.0,
IMSL_SIN,
omega,
IMSL_ERR_EST, &err_est,
IMSL_N_EVALS,
&n_evals,
0);
/* Print the result and the
*/
/* exact answer */
exact =
-.1281316;
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) ? 0.0 :
log(x);
}
Output
integral =
-0.128
exact =
-0.128
error estimate = 7.504603e-05
exact
error = 5.245209e-06
The number of function
evaluations = 215
Warning Errors
IMSL_ROUNDOFF_CONTAMINATION Roundoff error, preventing the requested tolerance from being achieved, has been detected.
IMSL_PRECISION_DEGRADATION A degradation in precision has been detected.
IMSL_EXTRAPOLATION_ROUNDOFF Roundoff error in the extrapolation table, preventing the requested tolerance from being achieved, has been detected.
Fatal Errors
IMSL_DIVERGENT Integral is probably divergent or slowly convergent.
IMSL_MAX_SUBINTERVALS The maximum number of subintervals allowed has been reached.
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