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_wset (d01tbc) or nag_quad_1d_gauss_wgen (d01tcc) 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.

4

References

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

5

Arguments

1: $ndim$ – IntegerInput

On entry:

n

, the number of dimensions of the integral.

Constraint:

1 \leq ndim \leq 20

2: $nptvec [ndim]$ – const IntegerInput

On entry:

nptvec [j - 1]

must specify the number of points in the

j

th dimension of the summation, for

j = 1, 2, \dots, n

3: $lwa$ – IntegerInput

On entry: the dimension of the arrays weight and abscis.

Constraint:

lwa \geq nptvec [0] + nptvec [1] + \dots + nptvec [ndim - 1]

4: $weight [lwa]$ – const doubleInput

On entry: must contain in succession the weights for the various dimensions, i.e.,

weight [k - 1]

contains the

i

th weight for the

j

th dimension, with

k = nptvec [0] + nptvec [1] + \dots + nptvec [j - 2] + i .

5: $abscis [lwa]$ – const doubleInput

On entry: must contain in succession the abscissae for the various dimensions, i.e.,

abscis [k - 1]

contains the

i

th abscissa for the

j

th dimension, with

k = nptvec [0] + nptvec [1] + \dots + nptvec [j - 2] + i .

6: $fun$ – function, supplied by the userExternal Function

fun must return the value of the integrand

f

at a specified point.

The specification of fun is:

double

fun (Integer ndim, const double x[], Nag_Comm *comm)

1: $ndim$ – IntegerInput

On entry:

n

, the number of dimensions of the integral.

2: $x [ndim]$ – const doubleInput

On entry: the coordinates of the point at which the integrand

f

must be evaluated.

3: $comm$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to fun.

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling nag_quad_md_gauss (d01fbc) you may allocate memory and initialize these pointers with various quantities for use by fun when called from nag_quad_md_gauss (d01fbc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: fun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_quad_md_gauss (d01fbc). If your code inadvertently does return any NaNs or infinities, nag_quad_md_gauss (d01fbc) is likely to produce unexpected results.

7: $comm$ – Nag_Comm *

The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

8: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $ndim = 〈value〉$ .
Constraint: $ndim \leq 20$ .

On entry, $ndim = 〈value〉$ .
Constraint: $ndim \geq 1$ .
NE_INT_2: On entry, lwa is too small. $lwa = 〈value〉$ . Minimum possible dimension: $〈value〉$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7

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.

8

Parallelism and Performance

nag_quad_md_gauss (d01fbc) is not threaded in any implementation.

9

Further Comments

The total time taken by nag_quad_md_gauss (d01fbc) will be proportional to

T \times nptvec [0] \times nptvec [1] \times \dots \times nptvec [ndim - 1],

where

T

is the time taken for one evaluation of fun.

10

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.