D01GBF returns an approximation to the integral of a function over a hyperrectangular region, using a Monte–Carlo method. An approximate relative error estimate is also returned. This routine is suitable for low accuracy work.
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: NDIM – INTEGERInput
On entry: $n$, the number of dimensions of the integral.
Constraint:
${\mathbf{NDIM}}\ge 1$.
 2: A(NDIM) – REAL (KIND=nag_wp) arrayInput
On entry: the lower limits of integration,
${a}_{i}$, for $\mathit{i}=1,2,\dots ,n$.
 3: B(NDIM) – REAL (KIND=nag_wp) arrayInput
On entry: the upper limits of integration,
${b}_{i}$, for $\mathit{i}=1,2,\dots ,n$.
 4: MINCLS – INTEGERInput/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 parameters 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: MAXCLS – INTEGERInput
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: FUNCTN – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure
FUNCTN must return the value of the integrand
$f$ at a given point.
The specification of
FUNCTN is:
FUNCTION FUNCTN ( 
NDIM, X) 
REAL (KIND=nag_wp) FUNCTN 
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.
FUNCTN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01GBF is called. Parameters denoted as
Input must
not be changed by this procedure.
 7: EPS – REAL (KIND=nag_wp)Input
On entry: the relative accuracy required.
Constraint:
${\mathbf{EPS}}\ge 0.0$.
 8: ACC – REAL (KIND=nag_wp)Output
On exit: the estimated relative accuracy of
FINEST.
 9: LENWRK – INTEGERInput
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: WRKSTR(LENWRK) – REAL (KIND=nag_wp) arrayInput/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: 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: 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).
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
A relative error estimate is output through the parameter
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.
The running time for D01GBF will usually be dominated by the time used to evaluate
FUNCTN, 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 D01GBF by calls to
G05SAF. Separate runs will produce identical answers.
This example calculates the integral
None.