NAG Library Routine Document
D01ATF
1 Purpose
D01ATF is a general purpose integrator which calculates an approximation to the integral of a function
$f\left(x\right)$ over a finite interval
$\left[a,b\right]$:
2 Specification
SUBROUTINE D01ATF ( 
F, A, B, EPSABS, EPSREL, RESULT, ABSERR, W, LW, IW, LIW, IFAIL) 
INTEGER 
LW, IW(LIW), LIW, IFAIL 
REAL (KIND=nag_wp) 
A, B, EPSABS, EPSREL, RESULT, ABSERR, W(LW) 
EXTERNAL 
F 

3 Description
D01ATF is based on the QUADPACK routine QAGS (see
Piessens et al. (1983)). It is an adaptive routine, 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 local error estimation is described in
Piessens et al. (1983).
The routine is suitable as a general purpose integrator, and can be used when the integrand has singularities, especially when these are of algebraic or logarithmic type.
D01ATF requires a subroutine to evaluate the integrand at an array of different points and is therefore amenable to parallel execution. Otherwise the algorithm is identical to that used by
D01AJF.
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 Arguments
 1: $\mathrm{F}$ – 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:
INTEGER 
N 
REAL (KIND=nag_wp) 
X(N), FV(N) 

 1: $\mathrm{X}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the points at which the integrand $f$ must be evaluated.
 2: $\mathrm{FV}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: ${\mathbf{FV}}\left(\mathit{j}\right)$ must contain the value of $f$ at the point ${\mathbf{X}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{N}}$.
 3: $\mathrm{N}$ – INTEGERInput

On entry: the number of points at which the integrand is to be evaluated. The actual value of
N is always
$21$.
F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01ATF is called. Arguments denoted as
Input must
not be changed by this procedure.
 2: $\mathrm{A}$ – REAL (KIND=nag_wp)Input

On entry: $a$, the lower limit of integration.
 3: $\mathrm{B}$ – REAL (KIND=nag_wp)Input

On entry: $b$, the upper limit of integration. It is not necessary that $a<b$.
 4: $\mathrm{EPSABS}$ – REAL (KIND=nag_wp)Input

On entry: the absolute accuracy required. If
EPSABS is negative, the absolute value is used. See
Section 7.
 5: $\mathrm{EPSREL}$ – REAL (KIND=nag_wp)Input

On entry: the relative accuracy required. If
EPSREL is negative, the absolute value is used. See
Section 7.
 6: $\mathrm{RESULT}$ – REAL (KIND=nag_wp)Output

On exit: the approximation to the integral $I$.
 7: $\mathrm{ABSERR}$ – REAL (KIND=nag_wp)Output

On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $\leftI{\mathbf{RESULT}}\right$.
 8: $\mathrm{W}\left({\mathbf{LW}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: details of the computation see
Section 9 for more information.
 9: $\mathrm{LW}$ – INTEGERInput

On entry: the dimension of the array
W as declared in the (sub)program from which D01ATF 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
${\mathbf{LW}}/4$. The more difficult the integrand, the larger
LW should be.
Suggested value:
${\mathbf{LW}}=800$ to $2000$ is adequate for most problems.
Constraint:
${\mathbf{LW}}\ge 4$.
 10: $\mathrm{IW}\left({\mathbf{LIW}}\right)$ – 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.
 11: $\mathrm{LIW}$ – INTEGERInput

On entry: the dimension of the array
IW as declared in the (sub)program from which D01ATF is called. The number of subintervals into which the interval of integration may be divided cannot exceed
LIW.
Suggested value:
${\mathbf{LIW}}={\mathbf{LW}}/4$.
Constraint:
${\mathbf{LIW}}\ge 1$.
 12: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation 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 arguments 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
${\mathbf{IFAIL}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Note: D01ATF 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.) you will probably gain from splitting up the interval at this point and calling the integrator on the subranges. 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.
 ${\mathbf{IFAIL}}=6$

On entry,  ${\mathbf{LW}}<4$, 
or  ${\mathbf{LIW}}<1$. 
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7 Accuracy
D01ATF 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
8 Parallelism and Performance
D01ATF is not threaded in any implementation.
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 D01ATF along with the integral contributions and error estimates over the 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 D01ATF 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
 ${r}_{i}={\mathbf{W}}\left(3n+i\right)$.
10 Example
10.1 Program Text
Program Text (d01atfe.f90)
10.2 Program Data
None.
10.3 Program Results
Program Results (d01atfe.r)