nag_quad_md_sphere (d01fdc) (PDF version)

d01 Chapter Contents

d01 Chapter Introduction

NAG Library Manual

NAG Library Function Document

nag_quad_md_sphere (d01fdc)

+− 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

nag_quad_md_sphere (d01fdc) calculates an approximation to a definite integral in up to

30

dimensions, using the method of Sag and Szekeres (see Sag and Szekeres (1964)). The region of integration is an

n

-sphere, or by built-in transformation via the unit

n

-cube, any product region.

2 Specification

#include <nag.h>

#include <nagd01.h>

void

nag_quad_md_sphere (Integer ndim,

double

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

double sigma,

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

Integer limit, double r0, double u, double *result, Integer *ncalls, Nag_Comm *comm, NagError *fail)

3 Description

nag_quad_md_sphere (d01fdc) 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 function 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

is

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 function 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}

is

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 – function, supplied by the userExternal Function

f must return the value of the integrand

f

at a given point.

The specification of f is:

double

f (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 f.

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling nag_quad_md_sphere (d01fdc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_quad_md_sphere (d01fdc) (see Section 3.2.1.1 in the Essential Introduction).

3: sigma – doubleInput

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 – function, supplied by the userExternal Function

If

sigma < 0.0

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

The specification of region is:

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

1: ndim – IntegerInput

On entry:

n

, the number of dimensions of the integral.

2: x[ndim] – const doubleInput

On entry:

x [0], \dots, x [j - 2]

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 – double *Output

On exit: the lower limit

c_{j}

of the range of

x_{j}

.

5: d – double *Output

On exit: the upper limit

d_{j}

of the range of

x_{j}

.

6: comm – Nag_Comm *

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

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling nag_quad_md_sphere (d01fdc) you may allocate memory and initialize these pointers with various quantities for use by region when called from nag_quad_md_sphere (d01fdc) (see Section 3.2.1.1 in the Essential Introduction).

If

sigma \geq 0.0

, region is not called by nag_quad_md_sphere (d01fdc), but the NAG defined null function pointer NULLFN must be supplied.

5: limit – IntegerInput

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

Constraint:

limit \geq 100

.

6: r0 – doubleInput

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 – doubleInput

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 – double *Output

On exit: the approximation to the integral

I

.

9: ncalls – Integer *Output

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

10: comm – Nag_Comm *Communication Structure

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

11: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $limit = ⟨value⟩$ .
Constraint: $limit \geq 100$ .
On entry, $ndim = ⟨value⟩$ .
Constraint: $ndim \leq 30$ .
On entry, $ndim = ⟨value⟩$ .
Constraint: $ndim \geq 1$ .
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.
NE_REAL: On entry, $r0 = ⟨value⟩$ .
Constraint: $r0 < 1.0$ .
On entry, $r0 = ⟨value⟩$ .
Constraint: $r0 > 0.0$ .
On entry, $u = ⟨value⟩$ .
Constraint: $u > 0.0$ .

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

Not applicable.

9 Further Comments

The time taken by nag_quad_md_sphere (d01fdc) 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,

or

\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 double 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 (d01fdce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (d01fdce.r)

nag_quad_md_sphere (d01fdc) (PDF version)

d01 Chapter Contents

d01 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014