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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_quad_withdraw_md_gauss (d01fb)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_quad_md_gauss (d01fb) computes an estimate of a multidimensional integral (from

1

20

dimensions), given the analytic form of the integrand and suitable Gaussian weights and abscissae.

Syntax

[result, ifail] = d01fb(nptvec, weight, abscis, fun, 'ndim', ndim, 'lwa', lwa)

[result, ifail] = nag_quad_withdraw_md_gauss(nptvec, weight, abscis, fun, 'ndim', ndim, 'lwa', lwa)

Description

nag_quad_md_gauss (d01fb) approximates a multidimensional integral by evaluating the summation

\sum_{i_{1} = 1}^{l_{1}} w_{1, i_{1}} \sum_{i_{2} = 1}^{l_{2}} w_{2, i_{2}} \dots \sum_{i_{n} = 1}^{l_{n}} w_{n, i_{n}} f (x_{1, i_{1}}, x_{2, i_{2}}, \dots, x_{n, i_{n}})

given the weights

w_{j, i_{j}}

and abscissae

x_{j, i_{j}}

for a multidimensional product integration rule (see Davis and Rabinowitz (1975)). The number of dimensions may be anything from

1

20

The weights and abscissae for each dimension must have been placed in successive segments of the arrays weight and abscis; for example, by calling nag_quad_1d_gauss_wgen (d01bc) or nag_quad_1d_gauss_wres (d01tb) once for each dimension using a quadrature formula and number of abscissae appropriate to the range of each

x_{j}

and to the functional dependence of

f

x_{j}

If normal weights are used, the summation will approximate the integral

\int w_{1} (x_{1}) \int w_{2} (x_{2}) \dots \int w_{n} (x_{n}) f (x_{1}, x_{2}, \dots, x_{n}) d x_{n} \dots d x_{2} d x_{1}

where

w_{j} (x)

is the weight function associated with the quadrature formula chosen for the

j

th dimension; while if adjusted weights are used, the summation will approximate the integral

\int \int \dots \int f (x_{1}, x_{2}, \dots, x_{n}) d x_{n} \dots d x_{2} d x_{1} .

You must supply a function to evaluate

f (x_{1}, x_{2}, \dots, x_{n})

at any values of

x_{1}, x_{2}, \dots, x_{n}

within the range of integration.

References

Davis P J and Rabinowitz P (1975) Methods of Numerical Integration Academic Press

Parameters

Compulsory Input Parameters

1: $nptvec (ndim)$ – int64int32nag_int array

nptvec (j)

must specify the number of points in the

j

th dimension of the summation, for

j = 1, 2, \dots, n

2: $weight (lwa)$ – double array

Must contain in succession the weights for the various dimensions, i.e.,

weight (k)

contains the

i

th weight for the

j

th dimension, with

k = nptvec (1) + nptvec (2) + \dots + nptvec (j - 1) + i .

3: $abscis (lwa)$ – double array

Must contain in succession the abscissae for the various dimensions, i.e.,

abscis (k)

contains the

i

th abscissa for the

j

th dimension, with

k = nptvec (1) + nptvec (2) + \dots + nptvec (j - 1) + i .

4: $fun$ – function handle or string containing name of m-file

fun must return the value of the integrand

f

at a specified point.

[result] = fun(ndim, x)

Input Parameters

1: $ndim$ – int64int32nag_int scalar: $n$ , the number of dimensions of the integral.
2: $x (ndim)$ – double array: The coordinates of the point at which the integrand $f$ must be evaluated.

Output Parameters

1: $result$ – double scalar: The value of $f (x)$ evaluated at x.

Optional Input Parameters

1: $ndim$ – int64int32nag_int scalar: Default: the dimension of the array nptvec.
$n$ , the number of dimensions of the integral.

Constraint: $1 \leq ndim \leq 20$ .
2: $lwa$ – int64int32nag_int scalar: Default: the dimension of the arrays weight, abscis. (An error is raised if these dimensions are not equal.)
The dimension of the arrays weight and abscis.

Constraint: $lwa \geq nptvec (1) + nptvec (2) + \dots + nptvec (ndim)$ .

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$ndim < 1$ ,
or	$ndim > 20$ ,
or	$lwa < nptvec (1) + nptvec (2) + \dots + nptvec (ndim)$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The accuracy of the computed multidimensional sum depends on the weights and the integrand values at the abscissae. If these numbers vary significantly in size and sign then considerable accuracy could be lost. If these numbers are all positive, then little accuracy will be lost in computing the sum.

Further Comments

The total time taken by nag_quad_md_gauss (d01fb) will be proportional to

T \times nptvec (1) \times nptvec (2) \times \dots \times nptvec (ndim),

where

T

is the time taken for one evaluation of fun.

Example

This example evaluates the integral

\int_{1}^{2} \int_{0}^{\infty} \int_{- \infty}^{\infty} \int_{1}^{\infty} \frac{{(x_{1} x_{2} x_{3})}^{6}}{{(x_{4} + 2)}^{8}} e^{- 2 x_{2}} e^{- 0.5 x_{3}^{2}} d x_{4} d x_{3} d x_{2} d x_{1}

using adjusted weights. The quadrature formulae chosen are:

$x_{1}$ : Gauss–Legendre, $a = 1.0$ , $b = 2.0$ ,
$x_{2}$ : Gauss–Laguerre, $a = 0.0$ , $b = 2.0$ ,
$x_{3}$ : Gauss–Hermite, $a = 0.0$ , $b = 0.5$ ,
$x_{4}$ : rational Gauss, $a = 1.0$ , $b = 2.0$ .

Four points are sufficient in each dimension, as this integral is in fact a product of four one-dimensional integrals, for each of which the chosen four-point formula is exact.

Open in the MATLAB editor: d01fb_example

function d01fb_example


fprintf('d01fb example results\n\n');

nptvec = [int64(4), 4, 4, 4];
a      = [1, 0, 0  , 1];
b      = [2, 2, 0.5, 2];
key    = [int64(0), -3, -4, -5];
j      = 1;
for i = 1:4
  [wgt, absc, ifail] = d01tb(key(i), a(i), b(i), nptvec(i));
  weight(j:j+nptvec(i)-1) = wgt;
  abscis(j:j+nptvec(i)-1) = absc;
  j = j + nptvec(i);
end

[result, ifail] = d01fb( ...
			 nptvec, weight, abscis, @fun);

fprintf('Result = %13.5f\n', result);



function result = fun(ndim,x)
  result = ((x(1)*x(2)*x(3))^6/(x(4)+2)^8)*exp(-2*x(2)-x(3)^2/2);

d01fb example results

Result =       0.25065

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Chapter Contents

Chapter Introduction

NAG Toolbox