.p>.CMCH4.DOC!INT_FCN_2D;int_fcn_2d
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), ¼, 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,
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, 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 the Introduction, Passing Data to
User-Supplied Functions at the beginning of 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 the Introduction, Passing Data to
User-Supplied Functions at the beginning of 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 the Introduction, Passing Data to
User-Supplied Functions at the beginning of 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 <math.h>
#include
<imsl.h>
float
fcn(float x, float y), gcn(float x), hcn(float
x);
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 <math.h>
#include
<imsl.h>
float
fcn(float x, float y), gcn(float x), hcn(float
x);
main()
{
int
n_evals;
float q,
exact, err_est,
exact_err;
/* 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.
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