int_fcn

CNLMath : Quadrature : int_fcn_2d

int_fcn_2d

Synopsis

Required Arguments

Return Value

Synopsis with Optional Arguments

Computes a two-dimensional iterated integral.

Synopsis

#include <imsl.h>

float imsl_f_int_fcn_2d (float fcn(), float a, float b, float gcn (float x), float hcn (float x), …, 0)

The type double function is imsl_d_int_fcn_2d.

Required Arguments

float fcn (float x, float y) (Input)
User-supplied function to be integrated.

float a (Input)
Lower limit of outer integral.

float b (Input)
Upper limit of outer integral.

float gcn (float x) (Input)
User-supplied function to evaluate the lower limit of the inner integral.

float hcn (float x) (Input)
User-supplied function to evaluate the upper limit of the inner integral.

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_2d (float fcn(), float a, float b, float gcn (), float hcn (),

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,

IMSL_GCN_W_DATA, float gcn(), void *data,

IMSL_HCN_W_DATA, float hcn(), void *data,

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, float y, 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.

IMSL_GCN_W_DATA, float gcn (float x, void *data), void *data (Input)
User supplied function to evaluate the lower limit of the inner integral, which also accepts a pointer to data that is supplied by the user. See Passing Data to User-Supplied Functions in the introduction to this manual for more details.

IMSL_HCN_W_DATA, float hcn (float x, void *data), void *data (Input)
User supplied function to evaluate the upper limit of the inner integral, 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_2d approximates the two-dimensional iterated integral

An estimate of the error is returned in err_est. The lower-numbered rules are used for less smooth integrands while the higher-order rules are more efficient for smooth (oscillatory) integrands.

Examples

Example 1

In this example, compute the value of the integral

#include <imsl.h>

#include <stdio.h>

#include <math.h>

float fcn(float x, float y), gcn(float x), hcn(float x);

int main()

{

float q, exact;

/* Evaluate the integral */

q = imsl_f_int_fcn_2d (fcn, 0.0, 1.0, gcn, hcn, 0);

/* print the result and the exact answer */

exact = 0.5*(cos(9.0)+cos(2.0)-cos(10.0)-cos(1.0));

printf("integral = %10.3f\nexact = %10.3f\n", q, exact);

}

float fcn(float x, float y)

{

return y * cos(x+y*y);

}

float gcn(float x)

{

return 1.0;

}

float hcn(float x)

{

return 3.0;

}

Output

integral = -0.514

exact = -0.514

Example 2

In this example, compute the value of the integral

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 is printed.

#include <imsl.h>

#include <stdio.h>

#include <math.h>

float fcn(float x, float y), gcn(float x), hcn(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_2d (fcn, 0., 1., gcn, hcn,

IMSL_ERR_EST, &err_est,

IMSL_N_EVALS, &n_evals,

0);

/* Print the result and the */

/* exact answer */

exact = 0.5*(cos(9.0)+cos(2.0)-cos(10.0)-cos(1.0));

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, float y)

{

return y * cos(x+y*y);

}

float gcn(float x)

{

return 1.0;

}

float hcn(float x)

{

return 3.0;

}

Output

integral = -0.514

exact = -0.514

error estimate = 3.065193e-06

exact error = 1.192093e-07

The number of function evaluations = 441

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 = "#".