NAG Library Routine Document
D01ALF
1 Purpose
D01ALF is a general purpose integrator which calculates an approximation to the integral of a function
$f\left(x\right)$ over a finite interval
$\left[a,b\right]$:
where the integrand may have local singular behaviour at a finite number of points within the integration interval.
2 Specification
SUBROUTINE D01ALF ( 
F, A, B, NPTS, POINTS, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) 
INTEGER 
NPTS, LW, IW(LIW), LIW, IFAIL 
REAL (KIND=nag_wp) 
F, A, B, POINTS(*), EPSABS, EPSREL, RESULT, ABSERR, W(LW) 
EXTERNAL 
F 

3 Description
D01ALF is based on the QUADPACK routine QAGP (see
Piessens et al. (1983)). It is very similar to
D01AJF, 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 usersupplied ‘break points’ always occur as the end points of some subinterval during the adaptive process. The local error estimation is described in
Piessens et al. (1983).
4 References
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
5 Parameters
 1: F – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure
F must return the value of the integrand
$f$ at a given point.
The specification of
F is:
 1: X – REAL (KIND=nag_wp)Input
On entry: 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 D01ALF is called. Parameters denoted as
Input must
not be changed by this procedure.
 2: A – REAL (KIND=nag_wp)Input
On entry: $a$, the lower limit of integration.
 3: B – REAL (KIND=nag_wp)Input
On entry: $b$, the upper limit of integration. It is not necessary that $a<b$.
 4: NPTS – INTEGERInput
On entry: the number of usersupplied break points within the integration interval.
Constraint:
${\mathbf{NPTS}}\ge 0$ and ${\mathbf{NPTS}}<\mathrm{min}\left(\left({\mathbf{LW}}2\times {\mathbf{NPTS}}4\right)/4,\left({\mathbf{LIW}}{\mathbf{NPTS}}2\right)/2\right)$.
 5: POINTS($*$) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array
POINTS
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NPTS}}\right)$.
On entry: the userspecified break points.
Constraint:
the break points must all lie within the interval of integration (but may be supplied in any order).
 6: EPSABS – REAL (KIND=nag_wp)Input
On entry: the absolute accuracy required. If
EPSABS is negative, the absolute value is used. See
Section 7.
 7: EPSREL – REAL (KIND=nag_wp)Input
On entry: the relative accuracy required. If
EPSREL is negative, the absolute value is used. See
Section 7.
 8: RESULT – REAL (KIND=nag_wp)Output
On exit: the approximation to the integral $I$.
 9: ABSERR – REAL (KIND=nag_wp)Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $\leftI{\mathbf{RESULT}}\right$.
 10: W(LW) – REAL (KIND=nag_wp) arrayOutput
On exit: details of the computation see
Section 8 for more information.
 11: LW – INTEGERInput
On entry: the dimension of the array
W as declared in the (sub)program from which D01ALF is called. The value of
LW (together with that of
LIW) imposes a bound on the number of subintervals into which the interval of integration may be divided by the routine. The number of subintervals cannot exceed
$\left({\mathbf{LW}}2\times {\mathbf{NPTS}}4\right)/4$. The more difficult the integrand, the larger
LW should be.
Suggested value:
a value in the range $800$ to $2000$ is adequate for most problems.
Constraint:
${\mathbf{LW}}\ge 2\times {\mathbf{NPTS}}+8$.
 12: IW(LIW) – INTEGER arrayOutput
On exit: ${\mathbf{IW}}\left(1\right)$ contains the actual number of subintervals used. The rest of the array is used as workspace.
 13: LIW – INTEGERInput
On entry: the dimension of the array
IW as declared in the (sub)program from which D01ALF is called. The number of subintervals into which the interval of integration may be divided cannot exceed
$\left({\mathbf{LIW}}{\mathbf{NPTS}}2\right)/2$.
Suggested value:
${\mathbf{LIW}}={\mathbf{LW}}/2$.
Constraint:
${\mathbf{LIW}}\ge {\mathbf{NPTS}}+4$.
 14: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ 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
${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is
$1$.
When the value $\mathbf{1}\text{ or}1$ 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).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Note: D01ALF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
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.
 ${\mathbf{IFAIL}}=2$
Roundoff error prevents the requested tolerance from being achieved. Consider requesting less accuracy.
 ${\mathbf{IFAIL}}=3$
Extremely bad local integrand behaviour causes a very strong subdivision around one (or more) points of the interval. The same advice applies as in the case of ${\mathbf{IFAIL}}={\mathbf{1}}$.
 ${\mathbf{IFAIL}}=4$
The requested tolerance cannot be achieved because the extrapolation does not increase the accuracy satisfactorily; the returned result is the best which 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. Please note that divergence can occur with any nonzero value of
IFAIL.
 ${\mathbf{IFAIL}}=6$
The input is invalid: break points are specified outside the integration range,
${\mathbf{NPTS}}>\mathrm{min}\left(\left({\mathbf{LW}}2\times {\mathbf{NPTS}}4\right)/4,\left({\mathbf{LIW}}{\mathbf{NPTS}}2\right)/2\right)$ or
${\mathbf{NPTS}}<0$.
RESULT and
ABSERR are set to zero.
 ${\mathbf{IFAIL}}=7$
On entry,  ${\mathbf{LW}}<2\times {\mathbf{NPTS}}+8$, 
or  ${\mathbf{LIW}}<{\mathbf{NPTS}}+4$. 
7 Accuracy
D01ALF cannot guarantee, but in practice usually achieves, the following accuracy:
where
and
EPSABS and
EPSREL are userspecified absolute and relative error tolerances. Moreover, it returns the quantity
ABSERR which, in normal circumstances, satisfies
The time taken by D01ALF depends on the integrand and the accuracy required.
If
${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, then you may wish to examine the contents of the array
W, which contains the end points of the subintervals used by D01ALF along with the integral contributions and error estimates over these subintervals.
Specifically, for
$i=1,2,\dots ,n$, let
${r}_{i}$ denote the approximation to the value of the integral over the subinterval
$\left[{a}_{i},{b}_{i}\right]$ 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
${\mathbf{RESULT}}={\displaystyle \sum _{i=1}^{n}}{r}_{i}$ unless D01ALF 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. The value of
$n$ is returned in
${\mathbf{IW}}\left(1\right)$,
and the values
${a}_{i}$,
${b}_{i}$,
${e}_{i}$ and
${r}_{i}$ are stored consecutively in the
array
W,
that is:
 ${a}_{i}={\mathbf{W}}\left(i\right)$,
 ${b}_{i}={\mathbf{W}}\left(n+i\right)$,
 ${e}_{i}={\mathbf{W}}\left(2n+i\right)$ and
 ${r}_{i}={\mathbf{W}}\left(3n+i\right)$.
9 Example
A break point is specified at $x=1/7$, at which point the integrand is infinite. (For definiteness the function FST returns the value $0.0$ at this point.)
9.1 Program Text
Program Text (d01alfe.f90)
9.2 Program Data
None.
9.3 Program Results
Program Results (d01alfe.r)