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Chapter Contents
Chapter Introduction
NAG Toolbox

## Purpose

nag_quad_1d_fin_osc (d01ak) is an adaptive integrator, especially suited to oscillating, nonsingular integrands, which calculates an approximation to the integral of a function $f\left(x\right)$ over a finite interval $\left[a,b\right]$:
 $I= ∫ab fx dx .$

## Syntax

[result, abserr, w, iw, ifail] = d01ak(f, a, b, epsabs, epsrel, 'lw', lw, 'liw', liw)
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_osc(f, a, b, epsabs, epsrel, 'lw', lw, 'liw', liw)

## Description

nag_quad_1d_fin_osc (d01ak) is based on the QUADPACK routine QAG (see Piessens et al. (1983)). It is an adaptive function, using the Gauss $30$-point and Kronrod $61$-point rules. A ‘global’ acceptance criterion (as defined by Malcolm and Simpson (1976)) is used. The local error estimation is described in Piessens et al. (1983).
Because nag_quad_1d_fin_osc (d01ak) is based on integration rules of high order, it is especially suitable for nonsingular oscillating integrands.
nag_quad_1d_fin_osc (d01ak) requires you to supply a function to evaluate the integrand at a single point.
The function nag_quad_1d_fin_osc_vec (d01au) uses an identical algorithm but requires you to supply a function to evaluate the integrand at an array of points. Therefore nag_quad_1d_fin_osc_vec (d01au) will be more efficient if the evaluation can be performed in vector mode on a vector-processing machine.

## References

Malcolm M A and Simpson R B (1976) Local versus global strategies for adaptive quadrature ACM Trans. Math. Software 1 129–146
Piessens R (1973) An algorithm for automatic integration Angew. Inf. 15 399–401
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag

## 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:     $\mathrm{epsabs}$ – double scalar
The absolute accuracy required. If epsabs is negative, the absolute value is used. See Accuracy.
5:     $\mathrm{epsrel}$ – double scalar
The relative accuracy required. If epsrel is negative, the absolute value is used. See Accuracy.

### Optional Input Parameters

1:     $\mathrm{lw}$int64int32nag_int scalar
Suggested value: ${\mathbf{lw}}=800$ to $2000$ is adequate for most problems.
Default: $800$
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 ${\mathbf{lw}}/4$. The more difficult the integrand, the larger lw should be.
Constraint: ${\mathbf{lw}}\ge 4$.
2:     $\mathrm{liw}$int64int32nag_int scalar
Default: ${\mathbf{lw}}/4$
The dimension of the array iw. the number of sub-intervals into which the interval of integration may be divided cannot exceed liw.
Constraint: ${\mathbf{liw}}\ge 1$.

### 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

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 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 ${\mathbf{ifail}}={\mathbf{1}}$.
${\mathbf{ifail}}=4$
 On entry, ${\mathbf{lw}}<4$, or ${\mathbf{liw}}<1$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
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_osc (d01ak) 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_osc (d01ak) 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_osc (d01ak) 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 ${\mathbf{result}}=\sum _{i=1}^{n}{r}_{i}$. 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
```function d01ak_example

fprintf('d01ak example results\n\n');

a = 0;
b = 2*pi;
epsabs = 0;
epsrel = 0.001;
f = @(x) x*(sin(30.0*x))*cos(x);
[result, abserr, w, iw, ifail] = d01ak(f, a, b, epsabs, epsrel);
fprintf('The approximation to the integral  = %10.6f\n',result);
fprintf('and the estimated absolute error   = %13.5e\n\n',abserr);

n = iw(1);
fprintf('The number of subintervals used = %d;\n',n);
fprintf('the limits of subintervals and their contributions are:\n\n');
fprintf(' subint   a_i      b_i       r_i\n');
for i= 1:n;
fprintf('%5d %8.4f %8.4f  %10.6f\n',i,w(i),w(i+n),w(i+3*n));
end

```
```d01ak example results

The approximation to the integral  =  -0.209672
and the estimated absolute error   =   4.47697e-14

The number of subintervals used = 4;
the limits of subintervals and their contributions are:

subint   a_i      b_i       r_i
1   0.0000   1.5708    0.000074
2   1.5708   3.1416    0.104762
3   3.1416   4.7124   -0.104910
4   4.7124   6.2832   -0.209598
```