NAG Library Routine Document

D01JAF

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

2 Specification

SUBROUTINE D01JAF (

F, NDIM, RADIUS, EPSA, EPSR, METHOD, ICOORD, RESULT, ESTERR, NEVALS, IFAIL)

INTEGER	NDIM, METHOD, ICOORD, NEVALS, IFAIL
REAL (KIND=nag_wp)	F, RADIUS, EPSA, EPSR, RESULT, ESTERR
EXTERNAL	F

3 Description

D01JAF 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 routine 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).

4 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

5 Parameters

1: 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:

FUNCTION F (

NDIM, X)

REAL (KIND=nag_wp) F

INTEGER	NDIM
REAL (KIND=nag_wp)	X(NDIM)

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. These coordinates are given in Cartesian or spherical polar form according to the value of ICOORD.

F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01JAF is called. Parameters denoted as Input must not be changed by this procedure.

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).

Note: D01JAF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

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

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

$IFAIL = 3$: The required accuracy cannot be achieved because the maximum accuracy with respect to the machine constants X02AJF and X02AMF has been attained. If this maximum accuracy is rather low (compared with X02AJF), 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

7 Accuracy

You can specify an absolute and/or a relative tolerance, setting EPSA and EPSR. The routine 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|

8 Further Comments

8.1 Timing

Timing depends on the integrand and the accuracy required.

8.2 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

8.3 Machine Dependencies

As a consequence of the transformation technique, the severity of the singularities which can be handled by D01JAF 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 D01JAF 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 D01JAF 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:
```
F = 1.0
A = X(1)
IF (A.GT.0.0) F = 1.0/(A*(2.0-A))**0.7
RETURN
```

Note that D01JAF ensures that

λ = X (1) > X02AMF

, but underflow could occur in the computation of

λ^{2}

9 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

9.1 Program Text

Program Text (d01jafe.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (d01jafe.r)

D01JAF (PDF version)

D01 Chapter Contents

D01 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentD01JAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

8.1 Timing

8.2 Spherical Polar Coordinates

8.3 Machine Dependencies

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

D01JAF