D01GBF returns an approximation to the integral of a function over a hyper-rectangular region, using a Monte Carlo method. An approximate relative error estimate is also returned. This routine is suitable for low accuracy work.
| SUBROUTINE D01GBF ( | NDIM, A, B, MINCLS, MAXCLS, FUNCTN, EPS, ACC, LENWRK, WRKSTR, FINEST, IFAIL) |
| INTEGER | NDIM, MINCLS, MAXCLS, LENWRK, IFAIL |
| double precision | A(NDIM), B(NDIM), FUNCTN, EPS, ACC, WRKSTR(LENWRK), FINEST |
| EXTERNAL | FUNCTN |
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×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 co-ordinate axis is determined and the routine uses this information to increase the density and change the widths of the sub-intervals 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.
Lautrup B (1971) An adaptive multi-dimensional integration procedure
Proc. 2nd Coll. Advanced Methods in Theoretical Physics, Marseille - 1: NDIM – INTEGERInput
On entry:
n, the number of dimensions of the integral.
Constraint:
NDIM≥1.
- 2: A(NDIM) – double precision arrayInput
On entry: the lower limits of integration, ai, for i=1,2,…,n.
- 3: B(NDIM) – double precision arrayInput
On entry: the upper limits of integration, bi, for i=1,2,…,n.
- 4: MINCLS – INTEGERInput/Output
On entry: must be set
- either to the minimum number of integrand evaluations to be allowed, in which case MINCLS≥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:
- MAXCLS>MINCLS;
- MAXCLS≥4×NDIM+1.
- 6: FUNCTN – double precision 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:
| double precision FUNCTION FUNCTN ( | NDIM, X) |
| INTEGER | NDIM |
| double precision | X(NDIM) |
- 1: NDIM – INTEGERInput
On entry: n, the number of dimensions of the integral.
- 2: X(NDIM) – double precision arrayInput
On entry: the co-ordinates of the point at which the integrand f must be evaluated.
FUNCTN must be 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 – double precisionInput
On entry: the relative accuracy required.
Constraint:
EPS≥0.0.
- 8: ACC – double precisionOutput
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×NDIM then the subroutine uses only one iteration of a crude Monte Carlo method with
MAXCLS sample points.
Constraint:
LENWRK≥10×NDIM.
- 10: WRKSTR(LENWRK) – double precision arrayInput/Output
On entry: if
MINCLS<0,
WRKSTR must be unchanged from the previous call of D01GBF – except that for a new integrand
WRKSTRLENWRK must be set to
0.0. See also
MINCLS.
On exit: contains information about the current sub-interval structure which could be used in later calls of D01GBF. In particular, WRKSTRj gives the number of sub-intervals used along the jth co-ordinate axis.
- 11: FINEST – double precisionInput/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 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 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
IFAIL≠0 on exit, the recommended value is
-1.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
If on entry
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 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.