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, ¼, 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 the Introduction, Passing Data to User-Supplied Functions at the beginning of 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.

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 <math.h>
#include <imsl.h>

float           fcn(float x);

main()
{
    float       q, exact;
                                /* 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 <math.h>
#include <imsl.h>

float           fcn(float x);
 
main()
{
    int         n_evals;
    float       q, exact, err_est, exact_err;
                                /* 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.

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