d01rlc is a general purpose integrator which calculates an approximation to the integral of a function $f\left(x\right)$ over a finite interval $[a,b]$:
The function may be called by the names: d01rlc or nag_quad_dim1_fin_brkpts.
3Description
d01rlc is based on the QUADPACK routine QAGP (see Piessens et al. (1983)). It is very similar to d01rjc, but allows you to supply ‘break-points’, points at which the integrand is known to be difficult. It employs an adaptive algorithm, using the Gauss $10$-point and Kronrod $21$-point rules. The algorithm, described in de Doncker (1978), incorporates a global acceptance criterion (as defined by Malcolm and Simpson (1976)) together with the $\epsilon $-algorithm (see Wynn (1956)) to perform extrapolation. The user-supplied ‘break-points’ always occur as the end points of some sub-interval during the adaptive process. The local error estimation is described in Piessens et al. (1983).
d01rlc requires you to supply a function to evaluate the integrand at an array of points.
4References
de Doncker E (1978) An adaptive extrapolation algorithm for automatic integration ACM SIGNUM Newsl.13(2) 12–18
Malcolm M A and Simpson R B (1976) Local versus global strategies for adaptive quadrature ACM Trans. Math. Software1 129–146
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag
Wynn P (1956) On a device for computing the ${e}_{m}\left({S}_{n}\right)$ transformation Math. Tables Aids Comput.10 91–96
5Arguments
1: $\mathbf{f}$ – function, supplied by the userExternal Function
f must return the values of the integrand $f$ at a set of points.
On entry: the abscissae,
${x}_{i}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nx}}$, at which function values are required.
2: $\mathbf{nx}$ – IntegerInput
On entry: the number of abscissae at which a function value is required. nx will be of size equal to the number of Kronrod points in the quadrature rule used, in this case $21$.
On exit: fv must contain the values of the integrand $f$. ${\mathbf{fv}}\left[i-1\right]=f\left({x}_{i}\right)$ for all $i=1,2,\dots ,{\mathbf{nx}}$.
4: $\mathbf{iflag}$ – Integer *Input/Output
On entry: ${\mathbf{iflag}}=0$.
On exit: set ${\mathbf{iflag}}<0$ to force an immediate exit with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_USER_STOP.
5: $\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling d01rlc you may allocate memory and initialize these pointers with various quantities for use by f when called from d01rlc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note:f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01rlc. If your code inadvertently does return any NaNs or infinities, d01rlc is likely to produce unexpected results.
2: $\mathbf{a}$ – doubleInput
On entry: $a$, the lower limit of integration.
3: $\mathbf{b}$ – doubleInput
On entry: $b$, the upper limit of integration. It is not necessary that $a<b$.
Note: if ${\mathbf{a}}={\mathbf{b}}$, the function will immediately return with ${\mathbf{result}}=0.0$, ${\mathbf{abserr}}=0.0$, ${\mathbf{rinfo}}=0.0$ and ${\mathbf{iinfo}}=0$.
4: $\mathbf{npts}$ – IntegerInput
On entry: the dimension of the array points. The number of user-supplied break-points within the integration interval.
Constraint:
the break-points must all lie within the interval of integration (but may be supplied in any order).
6: $\mathbf{epsabs}$ – doubleInput
On entry: ${\epsilon}_{a}$, the absolute accuracy required. If epsabs is negative, ${\epsilon}_{a}=\left|{\mathbf{epsabs}}\right|$. See Section 7.
7: $\mathbf{epsrel}$ – doubleInput
On entry: ${\epsilon}_{r}$, the relative accuracy required. If epsrel is negative, ${\epsilon}_{r}=\left|{\mathbf{epsrel}}\right|$. See Section 7.
8: $\mathbf{maxsub}$ – IntegerInput
On entry: ${\mathrm{max}}_{\mathit{sdiv}}$, the upper bound on the total number of subdivisions d01rlc may use to generate new segments. This does not include the initial segments of the interval generated from the supplied break-points. If ${\mathrm{max}}_{\mathit{sdiv}}=1$, only the initial segments will be evaluated.
Suggested value:
a value in the range $200$ to $500$ is adequate for most problems.
Constraint:
${\mathbf{maxsub}}\ge 1$.
9: $\mathbf{result}$ – double *Output
On exit: the approximation to the integral $I$.
10: $\mathbf{abserr}$ – double *Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $|I-{\mathbf{result}}|$.
Note: the dimension, dim, of the array rinfo
must be at least
$4\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{maxsub}},{\mathbf{npts}})+{\mathbf{npts}}+6$.
On exit: details of the computation. See Section 9 for more information.
Note: the dimension, dim, of the array iinfo
must be at least
$2\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{maxsub}},{\mathbf{npts}})+{\mathbf{npts}}+4$.
On exit: details of the computation. See Section 9 for more information.
13: $\mathbf{comm}$ – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
14: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{maxsub}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{maxsub}}\ge 1$.
On entry, ${\mathbf{npts}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{npts}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_QUAD_BAD_SUBDIV
Extremely bad integrand behaviour occurs around the sub-interval $(\u27e8\mathit{\text{value}}\u27e9,\u27e8\mathit{\text{value}}\u27e9)$. The same advice applies as in the case of ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_QUAD_MAX_SUBDIV.
NE_QUAD_BRKPTS_INVAL
On entry, break-points are specified outside $({\mathbf{a}},{\mathbf{b}})$: ${\mathbf{a}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{b}}=\u27e8\mathit{\text{value}}\u27e9$.
NE_QUAD_MAX_SUBDIV
The maximum number of subdivisions allowed with the given workspace has been reached without the accuracy requirements being achieved. Look at the integrand in order to determine the integration difficulties. If the position of a local difficulty within the interval can be determined (e.g., a singularity of the integrand or its derivative, a peak, a discontinuity, etc.) it should be supplied to the function as an element of the vector points. If necessary, another integrator, which is designed for handling the type of difficulty involved, must be used. Alternatively, consider relaxing the accuracy requirements specified by epsabs and epsrel, or increasing the amount of workspace.
NE_QUAD_NO_CONV
The integral is probably divergent or slowly convergent.
NE_QUAD_ROUNDOFF_EXTRAPL
Round-off error is detected in the extrapolation table. The requested tolerance cannot be achieved because the extrapolation does not increase the accuracy satisfactorily; the returned result is the best that can be obtained. The same advice applies as in the case of ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_QUAD_MAX_SUBDIV.
NE_QUAD_ROUNDOFF_TOL
Round-off error prevents the requested tolerance from being achieved: ${\mathbf{epsabs}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{epsrel}}=\u27e8\mathit{\text{value}}\u27e9$.
and epsabs and epsrel are user-specified absolute and relative error tolerances. Moreover, it returns the quantity abserr which, in normal circumstances, satisfies
The time taken by d01rlc depends on the integrand and the accuracy required.
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, NE_QUAD_BAD_SUBDIV, NE_QUAD_MAX_SUBDIV, NE_QUAD_NO_CONV, NE_QUAD_ROUNDOFF_EXTRAPL or NE_QUAD_ROUNDOFF_TOL, or if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_USER_STOP and at least one complete vector evaluation of f was completed, result and abserr will contain computed results.
If these results are unacceptable, or if otherwise required, then you may wish to examine the contents of the array rinfo, which contains the end points of the sub-intervals used by d01rlc along with the integral contributions and error estimates over the sub-intervals.
Note: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_USER_STOP, the results returned in result and abserr will only be
representative of the integral if ${n}_{\mathrm{seg}}$ is at least the
number of initial segments. That is ${n}_{\mathrm{seg}}\ge {\mathbf{npts}}+1$,
(assuming no repeated break-points have been supplied). If in doubt, you should examine the set of segment end points used, as described below,
and ensure these cover the interval $[a,b]$.
Specifically, for $i=1,2,\dots ,n$, let ${r}_{i}$ denote the approximation to the value of the integral over the sub-interval $[{a}_{i},{b}_{i}]$ in the partition of $[a,b]$ and ${e}_{i}$ be the corresponding absolute error estimate. Then, $\underset{{a}_{i}}{\overset{{b}_{i}}{\int}}}f\left(x\right)dx\simeq {r}_{i$ and ${\mathbf{result}}={\displaystyle \sum _{i=1}^{n}}{r}_{i}$, unless d01rlc terminates while testing for divergence of the integral (see Section 3.4.3 of Piessens et al. (1983)). In this case, result (and abserr) are taken to be the values returned from the extrapolation process. This will be indicated by ${\mathbf{iinfo}}\left[2\right]=1$ instead of $0$. The value of $n$ is returned in ${\mathbf{iinfo}}\left[0\right]$, and the values ${a}_{i}$, ${b}_{i}$, ${e}_{i}$ and ${r}_{i}$ are stored consecutively in the array rinfo, that is:
${a}_{i}={\mathbf{rinfo}}\left[i-1\right]$,
${b}_{i}={\mathbf{rinfo}}\left[n+i-1\right]$,
${e}_{i}={\mathbf{rinfo}}\left[2n+i-1\right]$ and
${r}_{i}={\mathbf{rinfo}}\left[3n+i-1\right]$.
The total number of abscissae at which the function was evaluated is returned in ${\mathbf{iinfo}}\left[1\right]$.
A break-point is specified at $x=1/7$, at which point the integrand is infinite. (For definiteness the function f returns the value $0.0$ at this point.)