Integer, Intent (In)	::	ndim, limit
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ncalls
Real (Kind=nag_wp), External	::	f
Real (Kind=nag_wp), Intent (In)	::	sigma, r0, u
Real (Kind=nag_wp), Intent (Out)	::	result
External	::	region

C Header Interface

#include nagmk26.h

void

d01fdf_ ( const Integer *ndim,
double (NAG_CALL *f)( const Integer *ndim, const double x[]),
const double *sigma,
void (NAG_CALL *region)( const Integer *ndim, const double x[], const Integer *j, double *c, double *d),
const Integer *limit, const double *r0, const double *u, double *result, Integer *ncalls, Integer *ifail)

3

Description

d01fdf calculates an approximation to

\int_{n -sphere of radius σ} f (x_{1}, x_{2}, \dots, x_{n}) d x_{1} d x_{2} \dots d x_{n}

(1)

or, more generally,

\int_{c_{1}}^{d_{1}} d x_{1} \dots \int_{c_{n}}^{d_{n}} d x_{n} f (x_{1}, \dots, x_{n})

(2)

where each

c_{i}

and

d_{i}

may be functions of

x_{j}

(j < i)

The routine uses the method of Sag and Szekeres (1964), which exploits a property of the shifted

p

-point trapezoidal rule, namely, that it integrates exactly all polynomials of degree

< p

(see Krylov (1962)). An attempt is made to induce periodicity in the integrand by making a parameterised transformation to the unit

n

-sphere. The Jacobian of the transformation and all its direct derivatives vanish rapidly towards the surface of the unit

n

-sphere, so that, except for functions which have strong singularities on the boundary, the resulting integrand will be pseudo-periodic. In addition, the variation in the integrand can be considerably reduced, causing the trapezoidal rule to perform well.

Integrals of the form (1) are transformed to the unit

n

-sphere by the change of variables:

x_{i} = y_{i} \frac{σ}{r} \tanh (\frac{u r}{1 - r^{2}})

where

r^{2} = \sum_{i = 1}^{n} y_{i}^{2}

and

u

is an adjustable parameter.

Integrals of the form (2) are first of all transformed to the

n

-cube

{[- 1, 1]}^{n}

by a linear change of variables

x_{i} = ((d_{i} + c_{i}) + (d_{i} - c_{i}) y_{i}) / 2

and then to the unit sphere by a further change of variables

y_{i} = \tanh (\frac{u z_{i}}{1 - r})

where

r^{2} = \sum_{i = 1}^{n} z_{i}^{2}

and

u

is again an adjustable parameter.

The parameter

u

in these transformations determines how the transformed integrand is distributed between the origin and the surface of the unit

n

-sphere. A typical value of

u

1.5

. For larger

u

, the integrand is concentrated toward the centre of the unit

n

-sphere, while for smaller

u

it is concentrated toward the perimeter.

In performing the integration over the unit

n

-sphere by the trapezoidal rule, a displaced equidistant grid of size

h

is constructed. The points of the mesh lie on concentric layers of radius

r_{i} = \frac{h}{4} \sqrt{n + 8 (i - 1)}, i = 1, 2, 3, \dots .

The routine requires you to specify an approximate maximum number of points to be used, and then computes the largest number of whole layers to be used, subject to an upper limit of

400

layers.

In practice, the rapidly-decreasing Jacobian makes it unnecessary to include the whole unit

n

-sphere and the integration region is limited by a user-specified cut-off radius

r_{0} < 1

. The grid-spacing

h

is determined by

r_{0}

and the number of layers to be used. A typical value of

r_{0}

0.8

Some experimentation may be required with the choice of

r_{0}

(which determines how much of the unit

n

-sphere is included) and

u

(which determines how the transformed integrand is distributed between the origin and surface of the unit

n

-sphere), to obtain best results for particular families of integrals. This matter is discussed further in Section 9.

4

References

Krylov V I (1962) Approximate Calculation of Integrals (trans A H Stroud) Macmillan

Sag T W and Szekeres G (1964) Numerical evaluation of high-dimensional integrals Math. Comput. 18 245–253

5

Arguments

1: $ndim$ – IntegerInput

On entry:

n

, the number of dimensions of the integral.

Constraint:

1 \leq ndim \leq 30

2: $f$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

f must return the value of the integrand

f

at a given point.

The specification of f is:

Fortran Interface

Function f (

ndim, x)

Real (Kind=nag_wp)	::	f
Integer, Intent (In)	::	ndim
Real (Kind=nag_wp), Intent (In)	::	x(ndim)

C Header Interface

#include nagmk26.h

double

f ( const Integer *ndim, const double x[])

1: $ndim$ – IntegerInput: On entry: $n$ , the number of dimensions of the integral.
2: $x (ndim)$ – Real (Kind=nag_wp) arrayInput: On entry: the coordinates of the point at which the integrand $f$ must be evaluated.

f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01fdf is called. Arguments denoted as Input must not be changed by this procedure.

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01fdf. If your code inadvertently does return any NaNs or infinities, d01fdf is likely to produce unexpected results.

3: $sigma$ – Real (Kind=nag_wp)Input

On entry: indicates the region of integration.

$sigma \geq 0.0$: The integration is carried out over the $n$ -sphere of radius sigma, centred at the origin.
$sigma < 0.0$: The integration is carried out over the product region described by region.

4: $region$ – Subroutine, supplied by the NAG Library or the user.External Procedure

sigma < 0.0

, region must evaluate the limits of integration in any dimension.

The specification of region is:

Fortran Interface

Subroutine region (

ndim, x, j, c, d)

Integer, Intent (In)	::	ndim, j
Real (Kind=nag_wp), Intent (In)	::	x(ndim)
Real (Kind=nag_wp), Intent (Out)	::	c, d

C Header Interface

#include nagmk26.h

void	region ( const Integer ndim, const double x[], const Integer j, double c, double d)

1: $ndim$ – IntegerInput: On entry: $n$ , the number of dimensions of the integral.
2: $x (ndim)$ – Real (Kind=nag_wp) arrayInput: On entry: $x (1), \dots, x (j - 1)$ contain the current values of the first $(j - 1)$ variables, which may be used if necessary in calculating $c_{j}$ and $d_{j}$ .
3: $j$ – IntegerInput: On entry: the index $j$ for which the limits of the range of integration are required.
4: $c$ – Real (Kind=nag_wp)Output: On exit: the lower limit $c_{j}$ of the range of $x_{j}$ .
5: $d$ – Real (Kind=nag_wp)Output: On exit: the upper limit $d_{j}$ of the range of $x_{j}$ .

region must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01fdf is called. Arguments denoted as Input must not be changed by this procedure.

Note: region should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01fdf. If your code inadvertently does return any NaNs or infinities, d01fdf is likely to produce unexpected results.

sigma \geq 0.0

, region is not called by d01fdf, but a dummy routine must be supplied (d01fdv may be used).

5: $limit$ – IntegerInput

On entry: the approximate maximum number of integrand evaluations to be used.

Constraint:

limit \geq 100

6: $r0$ – Real (Kind=nag_wp)Input

On entry: the cut-off radius on the unit

n

-sphere, which may be regarded as an adjustable parameter of the method.

Suggested value: a typical value is

r0 = 0.8

. (See also Section 9.)

Constraint:

0.0 < r0 < 1.0

7: $u$ – Real (Kind=nag_wp)Input

On entry: must specify an adjustable parameter of the transformation to the unit

n

-sphere.

Suggested value: a typical value is

u = 1.5

. (See also Section 9.)

Constraint:

u > 0.0

8: $result$ – Real (Kind=nag_wp)Output

On exit: the approximation to the integral

I

9: $ncalls$ – IntegerOutput

On exit: the actual number of integrand evaluations used. (See also Section 9.)

10: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this argument, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$

On entry,	$ndim < 1$ ,
or	$ndim > 30$ .

$ifail = 2$

On entry,

limit < 100

$ifail = 3$

On entry,	$r0 \leq 0.0$ ,
or	$r0 \geq 1.0$ .

$ifail = 4$

On entry,

u \leq 0.0

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

No error estimate is returned, but results may be verified by repeating with an increased value of limit (provided that this causes an increase in the returned value of ncalls).

8

Parallelism and Performance

d01fdf is not threaded in any implementation.

9

Further Comments

The time taken by d01fdf will be approximately proportional to the returned value of ncalls, which, except in the circumstances outlined in (b) below, will be close to the given value of limit.

(a)

Choice of

r_{0}

and

u

If the chosen combination of

r_{0}

and

u

is too large in relation to the machine accuracy it is possible that some of the points generated in the original region of integration may transform into points in the unit

n

-sphere which lie too close to the boundary surface to be distinguished from it to machine accuracy (despite the fact that

r_{0} < 1

). To be specific, the combination of

r_{0}

and

u

is too large if

\frac{u r_{0}}{1 - r_{0}^{2}} > 0.3465 (t - 1),   if ​ sigma \geq 0.0,

\frac{u r_{0}}{1 - r_{0}} > 0.3465 (t - 1),   if ​ sigma < 0.0,

where

t

is the number of bits in the mantissa of a real number.

The contribution of such points to the integral is neglected. This may be justified by appeal to the fact that the Jacobian of the transformation rapidly approaches zero towards the surface. Neglect of these points avoids the occurrence of overflow with integrands which are infinite on the boundary.

(b)

Values of limit and ncalls

limit is an approximate upper limit to the number of integrand evaluations, and may not be chosen less than

100

. There are two circumstances when the returned value of ncalls (the actual number of evaluations used) may be significantly less than limit.

Firstly, as explained in (a), an unsuitably large combination of

r_{0}

and

u

may result in some of the points being unusable. Such points are not included in the returned value of ncalls.

Secondly, no more than

400

layers will ever be used, no matter how high limit is set. This places an effective upper limit on ncalls as follows:

\begin{array}{lr} n = 1 : & 56 \\ n = 2 : & 1252 \\ n = 3 : & 23690 \\ n = 4 : & 394528 \\ n = 5 : & 5956906 \end{array}

10

Example

This example calculates the integral

\underset{s}{\int \int \int} \frac{d x_{1} d x_{2} d x_{3}}{\sqrt{σ^{2} - r^{2}}} = 22.2066

where

s

is the

3

-sphere of radius

σ

r^{2} = x_{1}^{2} + x_{2}^{2} + x_{3}^{2}

and

σ = 1.5

. Both sphere-to-sphere and general product region transformations are used. For the former, we use

r_{0} = 0.9

and

u = 1.5

; for the latter,

r_{0} = 0.8

and

u = 1.5

10.1

Program Text

Program Text (d01fdfe.f90)

10.2

Program Data

None.

10.3

Program Results

Program Results (d01fdfe.r)

d01fdf (PDF version)

D01 (quad) Chapter Contents

D01 (quad) Chapter Introduction

NAG Library Manual

NAG Library Routine Document

d01fdf (md_sphere)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results