NAG FL Interfaced01rlf (dim1_​fin_​brkpts)

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1Purpose

d01rlf is a general purpose integrator which calculates an approximation to the integral of a function $f\left(x\right)$ over a finite interval $\left[a,b\right]$:
 $I= ∫ab f(x) dx ,$
where the integrand may have local singular behaviour at a finite number of points within the integration interval.

2Specification

Fortran Interface
 Subroutine d01rlf ( f, a, b, npts,
 Integer, Intent (In) :: npts, maxsub Integer, Intent (Inout) :: iuser(*), ifail Integer, Intent (Out) :: iinfo(2*max(maxsub,npts)+npts+4) Real (Kind=nag_wp), Intent (In) :: a, b, points(npts), epsabs, epsrel Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: result, abserr, rinfo(4*max(maxsub,npts)+npts+6) Type (c_ptr), Intent (In) :: cpuser External :: f
#include <nag.h>
 void d01rlf_ (void (NAG_CALL *f)(const double x[], const Integer *nx, double fv[], Integer *iflag, Integer iuser[], double ruser[], void **cpuser),const double *a, const double *b, const Integer *npts, const double points[], const double *epsabs, const double *epsrel, const Integer *maxsub, double *result, double *abserr, double rinfo[], Integer iinfo[], Integer iuser[], double ruser[], void **cpuser, Integer *ifail)
The routine may be called by the names d01rlf or nagf_quad_dim1_fin_brkpts.

3Description

d01rlf is based on the QUADPACK routine QAGP (see Piessens et al. (1983)). It is very similar to d01rjf, 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).
d01rlf requires you to supply a (sub)routine 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. Software 1 129–146
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: $\mathbf{f}$Subroutine, supplied by the user. External Procedure
f must return the values of the integrand $f$ at a set of points.
The specification of f is:
Fortran Interface
 Subroutine f ( x, nx, fv,
 Integer, Intent (In) :: nx Integer, Intent (Inout) :: iflag, iuser(*) Real (Kind=nag_wp), Intent (In) :: x(nx) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: fv(nx) Type (c_ptr), Intent (In) :: cpuser
 void f (const double x[], const Integer *nx, double fv[], Integer *iflag, Integer iuser[], double ruser[], void **cpuser)
1: $\mathbf{x}\left({\mathbf{nx}}\right)$Real (Kind=nag_wp) array Input
On entry: the abscissae, ${x}_{i}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nx}}$, at which function values are required.
2: $\mathbf{nx}$Integer Input
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 $21$.
3: $\mathbf{fv}\left({\mathbf{nx}}\right)$Real (Kind=nag_wp) array Output
On exit: fv must contain the values of the integrand $f$. ${\mathbf{fv}}\left(i\right)=f\left({x}_{i}\right)$ 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 ${\mathbf{ifail}}=-{\mathbf{1}}$.
5: $\mathbf{iuser}\left(*\right)$Integer array User Workspace
6: $\mathbf{ruser}\left(*\right)$Real (Kind=nag_wp) array User Workspace
7: $\mathbf{cpuser}$Type (c_ptr) User Workspace
f is called with the arguments iuser, ruser and cpuser as supplied to d01rlf. You should use the arrays iuser and ruser, and the data handle cpuser to supply information to f.
f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01rlf 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 d01rlf. If your code inadvertently does return any NaNs or infinities, d01rlf is likely to produce unexpected results.
2: $\mathbf{a}$Real (Kind=nag_wp) Input
On entry: $a$, the lower limit of integration.
3: $\mathbf{b}$Real (Kind=nag_wp) Input
On entry: $b$, the upper limit of integration. It is not necessary that $a.
Note: if ${\mathbf{a}}={\mathbf{b}}$, the routine will immediately return with ${\mathbf{result}}=0.0$, ${\mathbf{abserr}}=0.0$, ${\mathbf{rinfo}}=0.0$ and ${\mathbf{iinfo}}=0$.
4: $\mathbf{npts}$Integer Input
On entry: the dimension of the array points as declared in the (sub)program from which d01rlf is called. The number of user-supplied break-points within the integration interval.
Constraint: ${\mathbf{npts}}\ge 0$.
5: $\mathbf{points}\left({\mathbf{npts}}\right)$Real (Kind=nag_wp) array Input
On entry: the user-specified break-points.
Constraint: the break-points must all lie within the interval of integration (but may be supplied in any order).
6: $\mathbf{epsabs}$Real (Kind=nag_wp) Input
On entry: ${\epsilon }_{a}$, the absolute accuracy required. If epsabs is negative, ${\epsilon }_{a}=|{\mathbf{epsabs}}|$. See Section 7.
7: $\mathbf{epsrel}$Real (Kind=nag_wp) Input
On entry: ${\epsilon }_{r}$, the relative accuracy required. If epsrel is negative, ${\epsilon }_{r}=|{\mathbf{epsrel}}|$. See Section 7.
8: $\mathbf{maxsub}$Integer Input
On entry: ${\mathrm{max}}_{\mathit{sdiv}}$, the upper bound on the total number of subdivisions d01rlf 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: $\mathbf{result}$Real (Kind=nag_wp) Output
On exit: the approximation to the integral $I$.
10: $\mathbf{abserr}$Real (Kind=nag_wp) Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $|I-{\mathbf{result}}|$.
11: $\mathbf{rinfo}\left(4×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{maxsub}},{\mathbf{npts}}\right)+{\mathbf{npts}}+6\right)$Real (Kind=nag_wp) array Output
On exit: details of the computation. See Section 9 for more information.
12: $\mathbf{iinfo}\left(2×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{maxsub}},{\mathbf{npts}}\right)+{\mathbf{npts}}+4\right)$Integer array Output
On exit: details of the computation. See Section 9 for more information.
13: $\mathbf{iuser}\left(*\right)$Integer array User Workspace
14: $\mathbf{ruser}\left(*\right)$Real (Kind=nag_wp) array User Workspace
15: $\mathbf{cpuser}$Type (c_ptr) User Workspace
iuser, ruser and cpuser are not used by d01rlf, but are passed directly to f and may be used to pass information to this routine. If you do not need to reference cpuser, it should be initialized to c_null_ptr.
16: $\mathbf{ifail}$Integer Input/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 d01rlf may return useful information.
${\mathbf{ifail}}=1$
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 routine 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.
${\mathbf{ifail}}=2$
Round-off error prevents the requested tolerance from being achieved: ${\mathbf{epsabs}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{epsrel}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=3$
Extremely bad integrand behaviour occurs around the sub-interval $\left(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right)$. The same advice applies as in the case of ${\mathbf{ifail}}={\mathbf{1}}$.
${\mathbf{ifail}}=4$
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 ${\mathbf{ifail}}={\mathbf{1}}$.
${\mathbf{ifail}}=5$
The integral is probably divergent or slowly convergent.
${\mathbf{ifail}}=41$
On entry, ${\mathbf{npts}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{npts}}\ge 0$.
${\mathbf{ifail}}=51$
On entry, break-points are specified outside $\left({\mathbf{a}},{\mathbf{b}}\right)$: ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{b}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=81$
On entry, ${\mathbf{maxsub}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{maxsub}}\ge 1$.
${\mathbf{ifail}}=-1$
Exit from f with ${\mathbf{iflag}}<0$.
${\mathbf{ifail}}=-99$
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

d01rlf cannot guarantee, but in practice usually achieves, the following accuracy:
 $|I-result|≤tol ,$
where
 $tol= max{|epsabs|,|epsrel|×|I|} ,$
and epsabs and epsrel are user-specified absolute and relative error tolerances. Moreover, it returns the quantity abserr which, in normal circumstances, satisfies
 $|I-result|≤abserr≤tol.$

8Parallelism and Performance

d01rlf is not threaded in any implementation.

The time taken by d01rlf depends on the integrand and the accuracy required.
If ${\mathbf{ifail}}={\mathbf{0}}$, ${\mathbf{1}}$, ${\mathbf{2}}$, ${\mathbf{3}}$, ${\mathbf{4}}$ or ${\mathbf{5}}$, or if ${\mathbf{ifail}}=-{\mathbf{1}}$ 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 d01rlf along with the integral contributions and error estimates over the sub-intervals.
Note: if ${\mathbf{ifail}}=-{\mathbf{1}}$, 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 $\left[a,b\right]$.
Specifically, for $i=1,2,\dots ,n$, let ${r}_{i}$ denote the approximation to the value of the integral over the sub-interval $\left[{a}_{i},{b}_{i}\right]$ in the partition of $\left[a,b\right]$ 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 ${\mathbf{result}}=\sum _{i=1}^{n}{r}_{i}$, unless d01rlf 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 ${\mathbf{iinfo}}\left(3\right)=1$ instead of $0$. The value of $n$ is returned in ${\mathbf{iinfo}}\left(1\right)$, and the values ${a}_{i}$, ${b}_{i}$, ${e}_{i}$ and ${r}_{i}$ are stored consecutively in the array rinfo, that is:
• ${a}_{i}={\mathbf{rinfo}}\left(i\right)$,
• ${b}_{i}={\mathbf{rinfo}}\left(n+i\right)$,
• ${e}_{i}={\mathbf{rinfo}}\left(2n+i\right)$ and
• ${r}_{i}={\mathbf{rinfo}}\left(3n+i\right)$.
The total number of abscissae at which the function was evaluated is returned in ${\mathbf{iinfo}}\left(2\right)$.

10Example

This example computes
 $∫ 0 1 1 |x-1/7| dx .$
A break-point is specified at $x=1/7$, at which point the integrand is infinite. (For definiteness the subroutine f returns the value $0.0$ at this point.)

10.1Program Text

Program Text (d01rlfe.f90)

None.

10.3Program Results

Program Results (d01rlfe.r)