IMSL C Math Library
int_fcn_cauchy

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Computes integrals of the form
in the Cauchy principal value sense.
Synopsis
#include <imsl.h>
float imsl_f_int_fcn_cauchy (float fcn(), float a, float b, float c, …, 0)
The type double function is imsl_d_int_fcn_cauchy.
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.
float c (Input)
Singular point, c must not equal a or b.
Return Value
The value of
is returned. If no value can be computed, NaN is returned.
Synopsis with Optional Arguments
#include <imsl.h>
float imsl_f_int_fcn_cauchy (float fcn(), float a, float b, float c,
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_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_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 Passing Data to User-Supplied Functions in the introduction to this manual for more details.
Description
The function imsl_f_int_fcn_cauchy uses a globally adaptive scheme in an attempt to reduce the absolute error. It computes integrals whose integrands have the special form w(x)f(x) where w(x) = 1∕(x − c). If c lies in the interval of integration, then the integral is interpreted as a Cauchy principal value. A combination of modified Clenshaw-Curtis and Gauss-Kronrod formulas are employed.
On some platforms, imsl_f_int_fcn_cauchy 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_cauchy is an implementation of the subroutine QAWC by Piessens et al. (1983).
Examples
Example 1
The Cauchy principal value of
is computed.
 
#include <imsl.h>
#include <stdio.h>
#include <math.h>
 
float fcn(float x);
 
int main()
{
float q, exact;
 
imsl_omp_options(
IMSL_SET_FUNCTIONS_THREAD_SAFE, 1,
0);
 
/* Evaluate the integral */
q = imsl_f_int_fcn_cauchy (fcn, -1.0, 5.0, 0.0, 0);
 
/* Print the result and the */
/* exact answer */
exact = log(125./631.)/18.;
printf("integral = %10.3f\nexact = %10.3f\n", q, exact);
}
 
float fcn(float x)
{
return 1.0/(5.0*x*x*x+6.0);
}
Output
 
integral = -0.090
exact = -0.090
Example 2
The Cauchy principal 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.
 
#include <imsl.h>
#include <stdio.h>
#include <math.h>
 
float fcn(float x);
 
int main()
{
int n_evals;
float q, exact, err_est, exact_err;
 
imsl_omp_options(
IMSL_SET_FUNCTIONS_THREAD_SAFE, 1,
0);
/* Evaluate the integral */
q = imsl_f_int_fcn_cauchy (fcn, -1.0, 5.0, 0.0,
IMSL_ERR_EST, &err_est,
IMSL_N_EVALS, &n_evals,
0);
 
/* Print the result and the */
/* exact answer */
exact = log(125./631.)/18.;
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 1.0/(5.0*x*x*x+6.0);
}
Output
 
integral = -0.090
exact = -0.090
error estimate = 2.160174e-06
exact error = 0.000000e+00
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.
Fatal Errors
IMSL_MAX_SUBINTERVALS
The maximum number of subintervals allowed has been reached.
IMSL_STOP_USER_FCN
Request from user supplied function to stop algorithm. User flag = "#".