), to a user-specified absolute or relative accuracy, by means of a modified Sag–Szekeres method. The function can handle singularities on the surface or at the centre of the sphere, and returns an error estimate.

Syntax

[result, esterr, nevals, ifail] = d01ja(f, ndim, radius, epsa, epsr, icoord, 'method', method)

[result, esterr, nevals, ifail] = nag_quad_md_sphere_bad(f, ndim, radius, epsa, epsr, icoord, 'method', method)

Description

nag_quad_md_sphere_bad (d01ja) calculates an approximation to the

n

-dimensional integral

I = \int \dots \int_{S} F (x_{1}, \dots, x_{n}) d x_{1} \dots d x_{n}, 2 \leq n \leq 4,

where

S

is the hypersphere

\sqrt{(x_{1}^{2} + \dots + x_{n}^{2})} \leq α < \infty

(the integrand function may also be defined in spherical coordinates). The algorithm is based on the Sag–Szekeres method (see Sag and Szekeres (1964)), applying the product trapezoidal formula after a suitable radial transformation. An improved transformation technique is developed: depending on the behaviour of the function and on the required accuracy, different transformations can be used, some of which are ‘double exponential’, as defined by Takahasi and Mori (1974). The resulting technique allows the function to deal with integrand singularities on the surface or at the centre of the sphere. When the estimated error of the approximation with mesh size

h

is larger than the tolerated error, the trapezoidal formula with mesh size

h / 2

is calculated. A drawback of this method is the exponential growth of the number of function evaluations in the successive approximations (this number grows with a factor

\approx 2^{n}

). This introduces the restriction

n \leq 4

. Because the convergence rate of the successive approximations is normally better than linear, the error estimate is based on the linear extrapolation of the difference between the successive approximations (see Robinson and de Doncker (1981) and Roose and de Doncker (1981)). For further details of the algorithm, see Roose and de Doncker (1981).

References

Robinson I and de Doncker E (1981) Automatic computation of improper integrals over a bounded or unbounded planar region Computing 27 89–284

Roose D and de Doncker E (1981) Automatic integration over a sphere J. Comput. Appl. Math. 7 203–224

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

Takahasi H and Mori M (1974) Double Exponential Formulas for Numerical Integration 9 Publ. RIMS, Kyoto University 721–741

Parameters

Compulsory Input Parameters

1: $f$ – function handle or string containing name of m-file

f must return the value of the integrand

f

at a given point.

[result] = f(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. These coordinates are given in Cartesian or spherical polar form according to the value of icoord.

Output Parameters

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

Optional Input Parameters

1: $method$ – int64int32nag_int scalar

Default:

0

Must specify the transformation to be used by the function. The choice depends on the behaviour of the integrand and on the required accuracy.

For well-behaved functions and functions with mild singularities on the surface of the sphere only:

$method = 1$: Low accuracy required.
$method = 2$: High accuracy required.

For functions with severe singularities on the surface of the sphere only:

$method = 3$: Low accuracy required.
$method = 4$: High accuracy required.
(in this case icoord must be set to $icoord = 2$ , and the function defined in special spherical coordinates).

For functions with a singularity at the centre of the sphere (and possibly with singularities on the surface as well):

$method = 5$: Low accuracy required.
$method = 6$: High accuracy required.

method = 0

can be used as a default value and is equivalent to:

$method = 1$ if $epsr > 10^{- 6}$ , and
$method = 2$ if $epsr \leq 10^{- 6}$ .

The distinction between low and high required accuracies, as mentioned above, depends also on the behaviour of the function. Roughly one may assume the critical value of epsa and epsr to be

10^{- 6}

, but the critical value will be smaller for a well-behaved integrand and larger for an integrand with severe singularities.

Constraint:

method = 0

1

2

3

4

5

6

icoord = 2

method = 3

4

Output Parameters

1: $result$ – double scalar: The approximation to the integral $I$ .
2: $esterr$ – double scalar: An estimate of the modulus of the absolute error.
3: $nevals$ – int64int32nag_int scalar: The number of function evaluations used.
4: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_quad_md_sphere_bad (d01ja) 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.

W $ifail = 1$: The required accuracy cannot be achieved within a limiting number of function evaluations (which is set by the function).

W $ifail = 2$: The required accuracy cannot be achieved because of round-off error.

W $ifail = 3$: The required accuracy cannot be achieved because the maximum accuracy with respect to the machine constants nag_machine_precision (x02aj) and nag_machine_real_safe (x02am) has been attained. If this maximum accuracy is rather low (compared with nag_machine_precision (x02aj)), the cause of the problem is a severe singularity on the boundary or at the centre of the sphere. If $method = 0$ , $1$ or $2$ , then setting $method = 3$ or $4$ may help.

$ifail = 4$

On entry,	$ndim < 2$ or $ndim > 4$ ,
or	$radius < 0.0$ ,
or	$method \neq 0$ , $1$ , $2$ , $3$ , $4$ , $5$ or $6$ ,
or	$icoord \neq 0$ , $1$ or $2$ ,
or	$icoord = 2$ and $method \neq 3$ or $4$ ,
or	$method = 3$ or $4$ and $icoord \neq 2$ .

No calculations have been performed. result and esterr are set to

0.0

$ifail = - 99$: 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

You can specify an absolute and/or a relative tolerance, setting epsa and epsr. The function attempts to calculate an approximation result such that

|I - result| \leq \max \{epsa, epsr \times |I|\} .

0 \leq ifail \leq 3

, esterr returns an estimate of, but not necessarily a bound for,

|I - result|

Further Comments

Timing

Timing depends on the integrand and the accuracy required.

Spherical Polar Coordinates

Cartesian coordinates are related to the spherical polar coordinates by:

\begin{array}{cl} x_{1} & = & ρ . \sin θ_{1} \dots \sin θ_{n - 2} . \sin θ_{n - 1} \\ x_{2} & = & ρ . \sin θ_{1} \dots \sin θ_{n - 2} . \cos θ_{n - 1} \\ x_{3} & = & ρ . \sin θ_{1} \dots \cos θ_{n - 2} \\ ⋮ \\ x_{n} & = & ρ . \cos θ_{1} \end{array}

where

0 < θ_{i} < π

, for

i = 1, 2, \dots, n - 2

and

0 < θ_{n - 1} < 2 π

Machine Dependencies

As a consequence of the transformation technique, the severity of the singularities which can be handled by nag_quad_md_sphere_bad (d01ja) depends on the precision and range of real numbers on the machine.

method = 3

4

must be used when the singularity on the surface is ‘severe’ in view of the requested accuracy and machine precision. In practice one has to set

method = 3

4

if nag_quad_md_sphere_bad (d01ja) terminates with

ifail = 3

when called with

method = 0

1

2

When integrating a function with a severe singular behaviour on the surface of the sphere, the additional transformation

ρ = α - λ

helps to avoid the loss of significant figures due to round-off error in the calculation of the integration nodes which are very close to the surface. For these points, the value of

λ

can be computed more accurately than the value of

ρ

. Naturally, care must be taken that the function subprogram does not contain expressions of the form

α - λ

, which could cause a large round-off error in the calculation of the integrand at the boundary of the sphere.

Care should be taken to avoid underflow and/or overflow problems in the function subprogram, because some of the integration nodes used by nag_quad_md_sphere_bad (d01ja) may be very close to the surface or to the centre of the sphere.

Example:

suppose the function
$f (ρ) = {(1 - ρ^{2})}^{- 0.7}$
is to be integrated over the unit sphere, with method=3 or 4. Then icoord should be set to 2; the transformation ρ=1-λ gives fρ=2λ-λ2 -0.7; and f could be coded thus:
```
function result = f(ndim, x)
  result = 1;
  a = x(1);
  if (a > 0)
    result = 1/(a*(2-a))^0.7;
  end
```

Note that nag_quad_md_sphere_bad (d01ja) ensures that

λ = x (1) > x02am

, but underflow could occur in the computation of

λ^{2}

Example

This example evaluates the integrals

\int \dots \int_{S} \frac{1}{\sqrt{1 - ρ^{2}}} d x_{1} \dots d x_{n}

where

ρ = \sqrt{\sum_{i = 1}^{n} x_{i}^{2}}

, and

S

is the unit sphere of dimension

n = 2 ​ or ​ 4

The exact values (to

12

decimal places) are

6.28318530718

and

13.1594725348

Open in the MATLAB editor: d01ja_example

function d01ja_example


fprintf('d01ja example results\n\n');

radius = 1;
epsa = 0;
epsr = 5e-05;
icoord = int64(1);

ndim = int64([2,4]);
for j = 1:2
  [result, esterr, nevals, ifail] = ...
  d01ja( ...
	 @f, ndim(j), radius, epsa, epsr, icoord);

  fprintf('Dimension of the sphere        = %8d\n', ndim(j));
  fprintf('Requested relative tolerance   = %8.5f\n',epsr);
  fprintf('Approximation to the integral  = %8.5f\n', result);
  fprintf('Number of function evaluations = %8d\n',nevals);
  fprintf('Estimated relative error       = %8.5f\n\n',esterr);
end



function result = f(ndim, x)
  a = (1-x(1))*(1+x(1));
  if (a == 0)
    result = 0;
  else
    result = 1/sqrt(a);
  end

d01ja example results

Dimension of the sphere        =        2
Requested relative tolerance   =  0.00005
Approximation to the integral  =  6.28319
Number of function evaluations =      193
Estimated relative error       =  0.00020

Dimension of the sphere        =        4
Requested relative tolerance   =  0.00005
Approximation to the integral  = 13.16004
Number of function evaluations =     2873
Estimated relative error       =  0.00053

PDF version (NAG web site, 64-bit version, 64-bit version)

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Chapter Introduction

NAG Toolbox