NAG Library Routine Document

D01JAF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{F}$ – 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:     $\mathrm{NDIM}$ – INTEGERInput

On entry: $n$, the number of dimensions of the integral.

2:     $\mathrm{X}\left(\mathbf{NDIM}\right)$ – 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.

See also Section 9.

2:     $\mathrm{NDIM}$ – INTEGERInput

On entry: $n$, the dimension of the sphere.

Constraint: $2\le \mathbf{NDIM}\le 4$.

3:     $\mathrm{RADIUS}$ – REAL (KIND=nag_wp)Input

On entry: $\alpha$, the radius of the sphere.

Constraint: $\mathbf{RADIUS}\ge 0.0$.

4:     $\mathrm{EPSA}$ – REAL (KIND=nag_wp)Input

On entry: the requested absolute tolerance. If $\mathbf{EPSA}<0.0$, its absolute value is used. See Section 7.

5:     $\mathrm{EPSR}$ – REAL (KIND=nag_wp)Input

On entry: the requested relative tolerance.

$\mathbf{EPSR}<0.0$

Its absolute value is used.

$\mathbf{EPSR}<10\times \left(\mathit{machine precision}\right)$

The latter value is used as EPSR by the routine. See Section 7.

6:     $\mathrm{METHOD}$ – INTEGERInput

On entry: must specify the transformation to be used by the routine. 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:

$\mathbf{METHOD}=1$

Low accuracy required.

$\mathbf{METHOD}=2$

High accuracy required.

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

$\mathbf{METHOD}=3$

Low accuracy required.

$\mathbf{METHOD}=4$

High accuracy required.

(in this case ICOORD must be set to $\mathbf{ICOORD}=2$, 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):

$\mathbf{METHOD}=5$

Low accuracy required.

$\mathbf{METHOD}=6$

High accuracy required.

$\mathbf{METHOD}=0$ can be used as a default value and is equivalent to:

• $\mathbf{METHOD}=1$ if $\mathbf{EPSR}>{10}^{-6}$, and
• $\mathbf{METHOD}=2$ if $\mathbf{EPSR}\le {10}^{-6}$.

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.

Suggested value: $\mathbf{METHOD}=0$.

Constraint: $\mathbf{METHOD}=0$, $1$, $2$, $3$, $4$, $5$ or $6$.

If $\mathbf{ICOORD}=2$, $\mathbf{METHOD}=3$ or $4$

7:     $\mathrm{ICOORD}$ – INTEGERInput

On entry: must specify which kind of coordinates are used in F.

$\mathbf{ICOORD}=0$

Cartesian coordinates $x_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.

$\mathbf{ICOORD}=1$

Spherical coordinates (see Section 9.2): $\mathbf{X}\left(1\right)=\rho$; $\mathbf{X}\left(\mathit{i}\right)={\theta }_{\mathit{i}-1}$, for $\mathit{i}=2,3,\dots ,n$.

$\mathbf{ICOORD}=2$,

Special spherical polar coordinates (see Section 9.3), with the additional transformation $\rho =\alpha -\lambda$: $\mathbf{X}\left(1\right)=\lambda =\alpha -\rho$; $\mathbf{X}\left(\mathit{i}\right)={\theta }_{\mathit{i}-1}$, for $\mathit{i}=2,3,\dots ,n$.

Constraint: $\mathbf{ICOORD}=0$, $1$ or $2$.

If $\mathbf{METHOD}=3$ or $4$, $\mathbf{ICOORD}=2$

8:     $\mathrm{RESULT}$ – REAL (KIND=nag_wp)Output

On exit: the approximation to the integral $I$.

9:     $\mathrm{ESTERR}$ – REAL (KIND=nag_wp)Output

On exit: an estimate of the modulus of the absolute error.

10:   $\mathrm{NEVALS}$ – INTEGEROutput

On exit: the number of function evaluations used.

11:   $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if $\mathbf{IFAIL}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

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

$\mathbf{IFAIL}=2$

The required accuracy cannot be achieved because of round-off error.

$\mathbf{IFAIL}=3$

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 $\mathbf{METHOD}=0$, $1$ or $2$, then setting $\mathbf{METHOD}=3$ or $4$ may help.

$\mathbf{IFAIL}=4$

On entry,	$\mathbf{NDIM}<2$ or $\mathbf{NDIM}>4$,
or	$\mathbf{RADIUS}<0.0$,
or	$\mathbf{METHOD}\ne 0$, $1$, $2$, $3$, $4$, $5$ or $6$,
or	$\mathbf{ICOORD}\ne 0$, $1$ or $2$,
or	$\mathbf{ICOORD}=2$ and $\mathbf{METHOD}\ne 3$ or $4$,
or	$\mathbf{METHOD}=3$ or $4$ and $\mathbf{ICOORD}\ne 2$.

No calculations have been performed. RESULT and ESTERR are set to $0.0$.

$\mathbf{IFAIL}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.8 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.7 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-999$

Dynamic memory allocation failed.

See Section 3.6 in the Essential Introduction for further information.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

9.1  Timing

9.2  Spherical Polar Coordinates

9.3  Machine Dependencies

• suppose the function

  $$f\left(\rho \right)={\left(1-{\rho }^2\right)}^{-0.7}$$

  is to be integrated over the unit sphere, with $\mathbf{METHOD}=3$ or $4$. Then ICOORD should be set to 2; the transformation $\rho =1-\lambda$ gives $f\left(\rho \right)={\left(2\lambda -{\lambda }^2\right)}^{-0.7}$; 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

10  Example

10.1  Program Text

Program Text (d01jafe.f90)

10.2  Program Data

10.3  Program Results

Program Results (d01jafe.r)