d01fcf attempts to evaluate a multidimensional integral (up to $15$ dimensions), with constant and finite limits, to a specified relative accuracy, using an adaptive subdivision strategy.
The routine may be called by the names d01fcf or nagf_quad_md_adapt.
3Description
d01fcf returns an estimate of a multidimensional integral over a hyper-rectangle (i.e., with constant limits), and also an estimate of the relative error. You set the relative accuracy required, return values for the integrand via a routine argument f, and also set the minimum and maximum acceptable number of calls to f (in minpts and maxpts).
The routine operates by repeated subdivision of the hyper-rectangular region into smaller hyper-rectangles. In each subregion, the integral is estimated using a seventh-degree rule, and an error estimate is obtained by comparison with a fifth-degree rule which uses a subset of the same points. The fourth differences of the integrand along each coordinate axis are evaluated, and the subregion is marked for possible future subdivision in half along that coordinate axis which has the largest absolute fourth difference.
If the estimated errors, totalled over the subregions, exceed the requested relative error (or if fewer than minpts calls to f have been made), further subdivision is necessary, and is performed on the subregion with the largest estimated error, that subregion being halved along the appropriate coordinate axis.
The routine will fail if the requested relative error level has not been attained by the time maxpts calls to f have been made; or, if the amount lenwrk of working storage is insufficient. A formula for the recommended value of lenwrk is given in Section 5. If a smaller value is used, and is exhausted in the course of execution, the routine switches to a less efficient mode of operation; only if this mode also breaks down is insufficient storage reported.
On entry: $n$, the number of dimensions of the integral.
2: $\mathbf{z}\left({\mathbf{ndim}}\right)$ – Real (Kind=nag_wp) arrayInput
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 d01fcf is called. Arguments denoted as Input must not be changed by this procedure.
Note:f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01fcf. If your code inadvertently does return any NaNs or infinities, d01fcf is likely to produce unexpected results.
7: $\mathbf{eps}$ – Real (Kind=nag_wp)Input
On entry: the relative error acceptable to you. When the solution is zero or very small relative accuracy may not be achievable but you may still set eps to a reasonable value and check for the error exit ${\mathbf{ifail}}={\mathbf{2}}$.
On entry: the dimension of the array wrkstr as declared in the (sub)program from which d01fcf is called.
Suggested value:
for maximum efficiency, ${\mathbf{lenwrk}}\ge ({\mathbf{ndim}}+2)\times (1+{\mathbf{maxpts}}/\alpha )$ (see argument maxpts for $\alpha $).
If lenwrk is less than this, d01fcf will usually run less efficiently and may fail.
10: $\mathbf{wrkstr}\left({\mathbf{lenwrk}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
11: $\mathbf{finval}$ – Real (Kind=nag_wp)Output
On exit: the best estimate obtained for the integral.
12: $\mathbf{ifail}$ – IntegerInput/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).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases d01fcf may return useful information.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{eps}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{eps}}>0.0$.
On entry, lenwrk is too small. ${\mathbf{lenwrk}}=\u27e8\mathit{\text{value}}\u27e9$. Minimum possible dimension: $\u27e8\mathit{\text{value}}\u27e9$.
On entry, maxpts is too small. ${\mathbf{maxpts}}=\u27e8\mathit{\text{value}}\u27e9$. Minimum possible dimension: $\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{minpts}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{maxpts}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{minpts}}\le {\mathbf{maxpts}}$.
On entry, ${\mathbf{ndim}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ndim}}\le 15$.
On entry, ${\mathbf{ndim}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ndim}}\ge 2$.
${\mathbf{ifail}}=2$
maxpts too small to obtain requested accuracy eps: ${\mathbf{maxpts}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{eps}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=3$
lenwrk was too small to complete computation. finval and acc contain estimates of integral and relative error, but acc is greater than eps.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
A relative error estimate is output through the argument acc.
8Parallelism and Performance
d01fcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d01fcf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
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 implementation-specific information.
9Further Comments
Execution time will usually be dominated by the time taken to evaluate f, and hence the maximum time that could be taken will be proportional to maxpts.