NAG Toolbox: nag_quad_md_adapt (d01fc)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathbf{ndim}\right)$ – double array

The lower limits of integration, $a_i$, for $\mathit{i}=1,2,\dots ,n$.

2:     $\mathrm{b}\left(\mathbf{ndim}\right)$ – double array

The upper limits of integration, $b_i$, for $\mathit{i}=1,2,\dots ,n$.

3:     $\mathrm{minpts}$ – int64int32nag_int scalar

Must be set to the minimum number of integrand evaluations to be allowed.

4:     $\mathrm{maxpts}$ – int64int32nag_int scalar

The maximum number of integrand evaluations to be allowed.

Constraints:

• $\mathbf{maxpts}\ge \mathbf{minpts}$;
• $\mathbf{maxpts}\ge \alpha$, where $\alpha =2^{\mathbf{ndim}}+2\times {\mathbf{ndim}}^2+2\times \mathbf{ndim}+1$.

5:     $\mathrm{functn}$ – function handle or string containing name of m-file

functn must return the value of the integrand $f$ at a given point.

[result] = functn(ndim, z)

Input Parameters

1:     $\mathrm{ndim}$ – int64int32nag_int scalar

$n$, the number of dimensions of the integral.

2:     $\mathrm{z}\left(\mathbf{ndim}\right)$ – double array

The coordinates of the point at which the integrand $f$ must be evaluated.

Output Parameters

1:     $\mathrm{result}$ – double scalar

The value of the integrand $f$ at the given point.

6:     $\mathrm{eps}$ – double scalar

The relative error acceptable to you. When the solution is zero or very small relative accuracy may not be achievable but you may still set eps to a reasonable value and check for the error exit $\mathbf{ifail}=\mathbf{2}$.

Constraint: $\mathbf{eps}>0.0$.

Optional Input Parameters

1:     $\mathrm{ndim}$ – int64int32nag_int scalar

Default: the dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)

$n$, the number of dimensions of the integral.

Constraint: $2\le \mathbf{ndim}\le 15$.

Output Parameters

1:     $\mathrm{minpts}$ – int64int32nag_int scalar

Contains the actual number of integrand evaluations used by nag_quad_md_adapt (d01fc).

2:     $\mathrm{acc}$ – double scalar

The estimated relative error in finval.

3:     $\mathrm{finval}$ – double scalar

The best estimate obtained for the integral.

4:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{ifail}=1$

On entry,	$\mathbf{ndim}<2$,
or	$\mathbf{ndim}>15$,
or	maxpts is too small,
or	$\mathit{lenwrk}<2\times \mathbf{ndim}+4$,
or	$\mathbf{eps}\le 0.0$.

W  $\mathbf{ifail}=2$

maxpts was too small to obtain the required relative accuracy eps. On soft failure, finval and acc contain estimates of the integral and the relative error, but acc will be greater than eps.

W  $\mathbf{ifail}=3$

lenwrk was too small. On soft failure, finval and acc contain estimates of the integral and the relative error, but acc will be greater than eps.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: d01fc_example

function d01fc_example


fprintf('d01fc example results\n\n');

a = zeros(4,1);
b = ones(4,1);
minpts  = int64(0);
maxpts  = int64(8000);
epsilon = 0.0001;

[minpts, acc, finval, ifail] = d01fc(...
                                     a, b, minpts, maxpts, @functn, epsilon);

fprintf('Requested accuracy = %10.2e\n', epsilon);
fprintf('Estimated value    = %8.4f\n',finval);
fprintf('Estimated accuracy = %9.1e\n',acc);



function result = functn(ndim,z)
  result = 4*z(1)*z(3)*z(3)*exp(2*z(1)*z(3))/(1+z(2)+z(4))^2;

d01fc example results

Requested accuracy =   1.00e-04
Estimated value    =   0.5754
Estimated accuracy =   9.9e-05