d01gbf uses an adaptive Monte Carlo method based on the algorithm described in
Lautrup (1971). It is implemented for integrals of the form:
Upon entry, unless
lenwrk has been set to the minimum value
$10\times {\mathbf{ndim}}$, the routine subdivides the integration region into a number of equal volume subregions. Inside each subregion the integral and the variance are estimated by means of pseudorandom sampling. All contributions are added together to produce an estimate for the whole integral and total variance. The variance along each coordinate axis is determined and the routine uses this information to increase the density and change the widths of the subintervals along each axis, so as to reduce the total variance. The total number of subregions is then increased by a factor of two and the program recycles for another iteration. The program stops when a desired accuracy has been reached or too many integral evaluations are needed for the next cycle.

1:
$\mathbf{ndim}$ – Integer
Input

On entry: $n$, the number of dimensions of the integral.
Constraint:
${\mathbf{ndim}}\ge 1$.

2:
$\mathbf{a}\left({\mathbf{ndim}}\right)$ – Real (Kind=nag_wp) array
Input

On entry: the lower limits of integration,
${a}_{i}$, for $\mathit{i}=1,2,\dots ,n$.

3:
$\mathbf{b}\left({\mathbf{ndim}}\right)$ – Real (Kind=nag_wp) array
Input

On entry: the upper limits of integration,
${b}_{i}$, for $\mathit{i}=1,2,\dots ,n$.

4:
$\mathbf{mincls}$ – Integer
Input/Output

On entry: must be set
 either to the minimum number of integrand evaluations to be allowed, in which case ${\mathbf{mincls}}\ge 0$;
 or to a negative value. In this case, the routine assumes that a previous call had been made with the same arguments ndim, a and b and with either the same integrand (in which case d01gbf continues calculation) or a similar integrand (in which case d01gbf begins the calculation with the subdivision used in the last iteration of the previous call). See also wrkstr.
On exit: contains the number of integrand evaluations actually used by d01gbf.

5:
$\mathbf{maxcls}$ – Integer
Input

On entry: the maximum number of integrand evaluations to be allowed. In the continuation case this is the number of new integrand evaluations to be allowed. These counts do not include zero integrand values.
Constraints:
 ${\mathbf{maxcls}}>{\mathbf{mincls}}$;
 ${\mathbf{maxcls}}\ge 4\times \left({\mathbf{ndim}}+1\right)$.

6:
$\mathbf{f}$ – real (Kind=nag_wp) Function, supplied by the user.
External Procedure

f must return the value of the integrand at a given point.
The specification of
f is:
Fortran Interface
Real (Kind=nag_wp) 
:: 
f 
Integer, Intent (In) 
:: 
ndim 
Real (Kind=nag_wp), Intent (In) 
:: 
x(ndim) 

C Header Interface
double 
f_ (const Integer *ndim, const double x[]) 

C++ Header Interface
#include <nag.h> extern "C" {
double 
f_ (const Integer &ndim, const double x[]) 
}


1:
$\mathbf{ndim}$ – Integer
Input

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

2:
$\mathbf{x}\left({\mathbf{ndim}}\right)$ – Real (Kind=nag_wp) array
Input

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
d01gbf is called. Arguments denoted as
Input must
not be changed by this procedure.
Note: f should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d01gbf. If your code inadvertently
does return any NaNs or infinities,
d01gbf is likely to produce unexpected results.

7:
$\mathbf{eps}$ – Real (Kind=nag_wp)
Input

On entry: the relative accuracy required.
Constraint:
${\mathbf{eps}}\ge 0.0$.

8:
$\mathbf{acc}$ – Real (Kind=nag_wp)
Output

On exit: the estimated relative accuracy of
finest.

9:
$\mathbf{lenwrk}$ – Integer
Input

On entry: the dimension of the array
wrkstr as declared in the (sub)program from which
d01gbf is called.
For maximum efficiency,
lenwrk should be about
If
lenwrk is given the value
$10\times {\mathbf{ndim}}$ then the subroutine uses only one iteration of a crude Monte Carlo method with
maxcls sample points.
Constraint:
${\mathbf{lenwrk}}\ge 10\times {\mathbf{ndim}}$.

10:
$\mathbf{wrkstr}\left({\mathbf{lenwrk}}\right)$ – Real (Kind=nag_wp) array
Input/Output

On entry: if
${\mathbf{mincls}}<0$,
wrkstr must be unchanged from the previous call of
d01gbf – except that for a new integrand
${\mathbf{wrkstr}}\left({\mathbf{lenwrk}}\right)$ must be set to
$0.0$. See also
mincls.
On exit: contains information about the current subinterval structure which could be used in later calls of d01gbf. In particular, ${\mathbf{wrkstr}}\left(j\right)$ gives the number of subintervals used along the $j$th coordinate axis.

11:
$\mathbf{finest}$ – Real (Kind=nag_wp)
Input/Output

On entry: must be unchanged from a previous call to d01gbf.
On exit: the best estimate obtained for the integral.

12:
$\mathbf{ifail}$ – Integer
Input/Output

On entry:
ifail must be set to
$0$,
$1$ or
$1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value
$1$ or
$1$ is recommended. If message printing is undesirable, then the value
$1$ is recommended. Otherwise, the value
$1$ is recommended since useful values can be provided in some output arguments even when
${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit.
When the value $\mathbf{1}$ or $\mathbf{1}$ 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).
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
A relative error estimate is output through the argument
acc. The confidence factor is set so that the actual error should be less than
acc 90% of the time. If you want a higher confidence level then a smaller value of
eps should be used.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The running time for
d01gbf will usually be dominated by the time used to evaluate the integrand
$f$, so the maximum time that could be used is approximately proportional to
maxcls.
For some integrands, particularly those that are poorly behaved in a small part of the integration region,
d01gbf may terminate with a value of
acc which is significantly smaller than the actual relative error. This should be suspected if the returned value of
mincls is small relative to the expected difficulty of the integral. Where this occurs,
d01gbf should be called again, but with a higher entry value of
mincls (e.g., twice the returned value) and the results compared with those from the previous call.
The exact values of
finest and
acc on return will depend (within statistical limits) on the sequence of random numbers generated within this routine.
This example calculates the integral
None.