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Chapter Contents
Chapter Introduction
NAG Toolbox

## Purpose

nag_quad_md_adapt (d01fc) attempts to evaluate a multidimensional integral (up to $15$ dimensions), with constant and finite limits, to a specified relative accuracy, using an adaptive subdivision strategy.

## Syntax

[minpts, acc, finval, ifail] = d01fc(a, b, minpts, maxpts, functn, eps, 'ndim', ndim)
[minpts, acc, finval, ifail] = nag_quad_md_adapt(a, b, minpts, maxpts, functn, eps, 'ndim', ndim)

## Description

nag_quad_md_adapt (d01fc) returns an estimate of a multidimensional integral over a hyper-rectangle (i.e., with constant limits), and also an estimate of the relative error. You set the relative accuracy required, return values for the integrand via a function argument functn, and also set the minimum and maximum acceptable number of calls to functn (in minpts and maxpts).
The function operates by repeated subdivision of the hyper-rectangular region into smaller hyper-rectangles. In each subregion, the integral is estimated using a seventh-degree rule, and an error estimate is obtained by comparison with a fifth-degree rule which uses a subset of the same points. The fourth differences of the integrand along each coordinate axis are evaluated, and the subregion is marked for possible future subdivision in half along that coordinate axis which has the largest absolute fourth difference.
If the estimated errors, totalled over the subregions, exceed the requested relative error (or if fewer than minpts calls to functn have been made), further subdivision is necessary, and is performed on the subregion with the largest estimated error, that subregion being halved along the appropriate coordinate axis.
The function will fail if the requested relative error level has not been attained by the time maxpts calls to functn have been made; or, if the amount lenwrk of working storage is insufficient. A formula for the recommended value of lenwrk is given in Arguments. If a smaller value is used, and is exhausted in the course of execution, the function switches to a less efficient mode of operation; only if this mode also breaks down is insufficient storage reported.
nag_quad_md_adapt (d01fc) is based on the HALF function developed by van Dooren and de Ridder (1976). It uses a different basic rule, described in Genz and Malik (1980).

## References

Genz A C and Malik A A (1980) An adaptive algorithm for numerical integration over an N-dimensional rectangular region J. Comput. Appl. Math. 6 295–302
van Dooren P and de Ridder L (1976) An adaptive algorithm for numerical integration over an N-dimensional cube J. Comput. Appl. Math. 2 207–217

## 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 $\alpha ={2}^{{\mathbf{ndim}}}+2×{{\mathbf{ndim}}}^{2}+2×{\mathbf{ndim}}+1$.
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
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

Note: nag_quad_md_adapt (d01fc) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

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 $\mathit{lenwrk}<2×{\mathbf{ndim}}+4$, 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.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

A relative error estimate is output through the argument acc.

Execution time will usually be dominated by the time taken to evaluate functn, and hence the maximum time that could be taken will be proportional to maxpts.

## Example

This example estimates the integral
 $∫01 ∫01 ∫01 ∫01 4 z1 z32 exp 2 z1 z3 1+ z2+ z4 2 dz4 dz3 dz2 dz1 = 0.575364 .$
The accuracy requested is one part in $10000$.
```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
```