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Chapter Contents
Chapter Introduction
NAG Toolbox

## Purpose

nag_quad_1d_fin_sing (d01al) is a general purpose integrator which calculates an approximation to the integral of a function $f\left(x\right)$ over a finite interval $\left[a,b\right]$:
 $I= ∫ab fx dx$
where the integrand may have local singular behaviour at a finite number of points within the integration interval.

## Syntax

[result, abserr, w, iw, ifail] = d01al(f, a, b, points, epsabs, epsrel, 'npts', npts, 'lw', lw, 'liw', liw)
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_sing(f, a, b, points, epsabs, epsrel, 'npts', npts, 'lw', lw, 'liw', liw)

## Description

nag_quad_1d_fin_sing (d01al) is based on the QUADPACK routine QAGP (see Piessens et al. (1983)). It is very similar to nag_quad_1d_fin_bad (d01aj), 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).

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{f}$ – function handle or string containing name of m-file
f must return the value of the integrand $f$ at a given point.
[result] = f(x)

Input Parameters

1:     $\mathrm{x}$ – double scalar
The point at which the integrand $f$ must be evaluated.

Output Parameters

1:     $\mathrm{result}$ – double scalar
The value of the integrand at x
2:     $\mathrm{a}$ – double scalar
$a$, the lower limit of integration.
3:     $\mathrm{b}$ – double scalar
$b$, the upper limit of integration. It is not necessary that $a.
4:     $\mathrm{points}\left(:\right)$ – double array
The dimension of the array points must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{npts}}\right)$
The user-specified break-points.
Constraint: the break-points must all lie within the interval of integration (but may be supplied in any order).
5:     $\mathrm{epsabs}$ – double scalar
The absolute accuracy required. If epsabs is negative, the absolute value is used. See Accuracy.
6:     $\mathrm{epsrel}$ – double scalar
The relative accuracy required. If epsrel is negative, the absolute value is used. See Accuracy.

### Optional Input Parameters

1:     $\mathrm{npts}$int64int32nag_int scalar
Default: the dimension of the array points.
The number of user-supplied break-points within the integration interval.
Constraint: ${\mathbf{npts}}\ge 0$ and ${\mathbf{npts}}<\mathrm{min}\left(\left({\mathbf{lw}}-2×{\mathbf{npts}}-4\right)/4,\left({\mathbf{liw}}-{\mathbf{npts}}-2\right)/2\right)$.
2:     $\mathrm{lw}$int64int32nag_int scalar
Default: a value in the range $800$ to $2000$ is adequate for most problems.
The dimension of the array w. the value of lw (together with that of liw) imposes a bound on the number of sub-intervals into which the interval of integration may be divided by the function. The number of sub-intervals cannot exceed $\left({\mathbf{lw}}-2×{\mathbf{npts}}-4\right)/4$. The more difficult the integrand, the larger lw should be.
Constraint: ${\mathbf{lw}}\ge 2×{\mathbf{npts}}+8$.
3:     $\mathrm{liw}$int64int32nag_int scalar
Default: ${\mathbf{lw}}/2$
The dimension of the array iw. the number of sub-intervals into which the interval of integration may be divided cannot exceed $\left({\mathbf{liw}}-{\mathbf{npts}}-2\right)/2$.
Constraint: ${\mathbf{liw}}\ge {\mathbf{npts}}+4$.

### Output Parameters

1:     $\mathrm{result}$ – double scalar
The approximation to the integral $I$.
2:     $\mathrm{abserr}$ – double scalar
An estimate of the modulus of the absolute error, which should be an upper bound for $\left|I-{\mathbf{result}}\right|$.
3:     $\mathrm{w}\left({\mathbf{lw}}\right)$ – double array
4:     $\mathrm{iw}\left({\mathbf{liw}}\right)$int64int32nag_int array
${\mathbf{iw}}\left(1\right)$ contains the actual number of sub-intervals used. The rest of the array is used as workspace.
5:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Note: nag_quad_1d_fin_sing (d01al) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ${\mathbf{ifail}}=1$
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 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.
W  ${\mathbf{ifail}}=2$
Round-off error prevents the requested tolerance from being achieved. Consider requesting less accuracy.
W  ${\mathbf{ifail}}=3$
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 ${\mathbf{ifail}}={\mathbf{1}}$.
W  ${\mathbf{ifail}}=4$
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 ${\mathbf{ifail}}={\mathbf{1}}$.
W  ${\mathbf{ifail}}=5$
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, ${\mathbf{npts}}>\mathrm{min}\left(\left({\mathbf{lw}}-2×{\mathbf{npts}}-4\right)/4,\left({\mathbf{liw}}-{\mathbf{npts}}-2\right)/2\right)$ or ${\mathbf{npts}}<0$. result and abserr are set to zero.
${\mathbf{ifail}}=7$
 On entry, ${\mathbf{lw}}<2×{\mathbf{npts}}+8$, or ${\mathbf{liw}}<{\mathbf{npts}}+4$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

nag_quad_1d_fin_sing (d01al) cannot guarantee, but in practice usually achieves, the following accuracy:
 $I-result≤tol,$
where
 $tol= maxepsabs,epsrel×I ,$
and epsabs and epsrel are user-specified absolute and relative error tolerances. Moreover, it returns the quantity abserr which, in normal circumstances, satisfies
 $I-result≤abserr≤tol.$

The time taken by nag_quad_1d_fin_sing (d01al) 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 sub-intervals used by nag_quad_1d_fin_sing (d01al) along with the integral contributions and error estimates over these sub-intervals.
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 nag_quad_1d_fin_sing (d01al) 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
• ${r}_{i}={\mathbf{w}}\left(3n+i\right)$.

## Example

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 function FST returns the value $0.0$ at this point.)
```function d01al_example

fprintf('d01al example results\n\n');

a = 0;
b = 1;
points = [1/7];
epsabs = 0;
epsrel = 0.001;
[result, abserr, w, iw, ifail] = ...
d01al( ...
@f, a, b, points, epsabs, epsrel);

fprintf('Result = %13.5f,  Standard error = %10.2e\n', result, abserr);

function [result] = f(x)
a = abs(x-1/7);
if (a ~= 0)
result = a^(-0.5);
else
result = 0;
end
```
```d01al example results

Result =       2.60757,  Standard error =   6.39e-14
```