nag_multid_quad_monte_carlo (d01gbc) evaluates 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 function is suitable for low accuracy work.
nag_multid_quad_monte_carlo (d01gbc) uses an adaptive Monte–Carlo method based on the algorithm described by
Lautrup (1971). It is implemented for integrals of the form:
Upon entry, unless the argument
${\mathbf{method}}=\mathrm{Nag\_OneIteration}$, the function 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 function 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: the number of dimensions of the integral, $n$.
Constraint:
${\mathbf{ndim}}\ge 1$.
 2:
f – function, supplied by the userExternal Function

f must return the value of the integrand
$f$ at a given point.
The specification of
f is:
double 
f (Integer ndim,
const double x[])


 1:
ndim – IntegerInput

On entry: the number of dimensions of the integral.
 2:
x[ndim] – const doubleInput

On entry: the coordinates of the point at which the integrand must be evaluated.
 3:
method – Nag_MCMethodInput
On entry: the method to be used.
 ${\mathbf{method}}=\mathrm{Nag\_OneIteration}$
 The function uses only one iteration of a crude Monte–Carlo method with maxcls sample points.
 ${\mathbf{method}}=\mathrm{Nag\_ManyIterations}$
 The function subdivides the integration region into a number of equal volume subregions.
Constraint:
${\mathbf{method}}=\mathrm{Nag\_OneIteration}$ or $\mathrm{Nag\_ManyIterations}$.
 4:
cont – Nag_StartInput
On entry: the continuation state of the evaluation of the integrand.
 ${\mathbf{cont}}=\mathrm{Nag\_Cold}$
 Indicates that this is the first call to the function with the current integrand and arguments ndim, a and b.
 ${\mathbf{cont}}=\mathrm{Nag\_Hot}$
 Indicates that a previous call has been made with the same arguments ndim, a and b with the same integrand. Please note that method must not be changed.
 ${\mathbf{cont}}=\mathrm{Nag\_Warm}$
 Indicates that a previous call has been made with the same arguments ndim, a and b but that the integrand is new. Please note that method must not be changed.
Constraint:
${\mathbf{cont}}=\mathrm{Nag\_Cold}$, $\mathrm{Nag\_Warm}$ or $\mathrm{Nag\_Hot}$.
 5:
a[ndim] – const doubleInput

On entry: the lower limits of integration, ${a}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
 6:
b[ndim] – const doubleInput

On entry: the upper limits of integration, ${b}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
 7:
mincls – Integer *Input/Output

On entry:
mincls must be set to the minimum number of integrand evaluations to be allowed.
Constraint:
$0\le {\mathbf{mincls}}<{\mathbf{maxcls}}$.
On exit:
mincls contains the total number of integrand evaluations actually used by nag_multid_quad_monte_carlo (d01gbc).
 8:
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:
 ${\mathbf{maxcls}}>{\mathbf{mincls}}$;
 ${\mathbf{maxcls}}\ge 4\times \left({\mathbf{ndim}}+1\right)$.
 9:
eps – doubleInput

On entry: the relative accuracy required.
Constraint:
${\mathbf{eps}}\ge 0.0$.
 10:
finest – double *Output

On exit: the best estimate obtained for the integral.
 11:
acc – double *Output

On exit: the estimated relative accuracy of
finest.
 12:
comm_arr – double **Input/Output

On entry: if
${\mathbf{cont}}=\mathrm{Nag\_Warm}$ or
$\mathrm{Nag\_Hot}$, the memory pointed to and allocated by a previous call of nag_multid_quad_monte_carlo (d01gbc) must be unchanged.
If ${\mathbf{cont}}=\mathrm{Nag\_Cold}$ then appropriate memory is allocated internally by nag_multid_quad_monte_carlo (d01gbc).
On exit:
comm_arr contains information about the current subinterval structure which could be used in later calls of nag_multid_quad_monte_carlo (d01gbc). In particular,
${\mathbf{comm\_arr}}\left[j1\right]$ gives the number of subintervals used along the
$j$th coordinate axis.
When this information is no longer useful, or before a subsequent call to nag_multid_quad_monte_carlo (d01gbc) with
${\mathbf{cont}}=\mathrm{Nag\_Cold}$ is made, you should free the storage contained in this pointer using the NAG macro
NAG_FREE. Note this memory will have been allocated and needs to be freed only if the error exit NE_NOERROR or
NE_QUAD_MAX_INTEGRAND_EVAL occurs. Otherwise, no memory needs to be freed.
 13:
fail – NagError *Input/Output

The NAG error argument (see
Section 3.6 in the Essential Introduction).
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 desire a higher confidence level then a smaller value of
eps should be used.
The running time for nag_multid_quad_monte_carlo (d01gbc) 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, this function 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, nag_multid_quad_monte_carlo (d01gbc) 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 function by calls to
nag_random_continuous_uniform (g05cac). Separate runs will produce identical answers unless the part of the program executed prior to calling this function also calls (directly or indirectly) functions from
Chapter g05, and the series of such calls differs between runs. If desired, you may ensure the identity or difference between runs of the results returned by this function, by calling
nag_random_init_repeatable (g05cbc) or
nag_random_init_nonrepeatable (g05ccc) respectively, immediately before calling this function.
This example calculates the integral
None.