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Chapter Contents
Chapter Introduction
NAG Toolbox

## Purpose

nag_quad_1d_fin_wcauchy (d01aq) calculates an approximation to the Hilbert transform of a function g(x)$g\left(x\right)$ over [a,b]$\left[a,b\right]$:
 b I = ∫ (g(x))/(x − c)dx a
$I=∫abg(x) x-c dx$
for user-specified values of a$a$, b$b$ and c$c$.

## Syntax

[result, abserr, w, iw, ifail] = d01aq(g, a, b, c, epsabs, epsrel, 'lw', lw, 'liw', liw)
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_wcauchy(g, a, b, c, epsabs, epsrel, 'lw', lw, 'liw', liw)

## Description

nag_quad_1d_fin_wcauchy (d01aq) is based on the QUADPACK routine QAWC (see Piessens et al. (1983)) and integrates a function of the form g(x)w(x)$g\left(x\right)w\left(x\right)$, where the weight function
 w(x) = 1/(x − c) $w(x)=1x-c$
is that of the Hilbert transform. (If a < c < b$a the integral has to be interpreted in the sense of a Cauchy principal value.) It is an adaptive function which employs a ‘global’ acceptance criterion (as defined by Malcolm and Simpson (1976)). Special care is taken to ensure that c$c$ is never the end point of a sub-interval (see Piessens et al. (1976)). On each sub-interval (c1,c2)$\left({c}_{1},{c}_{2}\right)$ modified Clenshaw–Curtis integration of orders 12$12$ and 24$24$ is performed if c1dcc2 + d${c}_{1}-d\le c\le {c}_{2}+d$ where d = (c2c1) / 20$d=\left({c}_{2}-{c}_{1}\right)/20$. Otherwise the Gauss
7$7$-point and Kronrod 15$15$-point rules are used. The local error estimation is described by
Piessens et al. (1983).

## 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, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag
Piessens R, van Roy–Branders M and Mertens I (1976) The automatic evaluation of Cauchy principal value integrals Angew. Inf. 18 31–35

## Parameters

### Compulsory Input Parameters

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

Input Parameters

1:     x – double scalar
The point at which the function g$g$ must be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.
2:     a – double scalar
a$a$, the lower limit of integration.
3:     b – double scalar
b$b$, the upper limit of integration. It is not necessary that a < b$a.
4:     c – double scalar
The parameter c$c$ in the weight function.
Constraint: c${\mathbf{c}}$ must not equal a or b.
5:     epsabs – double scalar
The absolute accuracy required. If epsabs is negative, the absolute value is used. See Section [Accuracy].
6:     epsrel – double scalar
The relative accuracy required. If epsrel is negative, the absolute value is used. See Section [Accuracy].

### Optional Input Parameters

1:     lw – int64int32nag_int scalar
The dimension of the array w as declared in the (sub)program from which nag_quad_1d_fin_wcauchy (d01aq) is called. 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 lw / 4${\mathbf{lw}}/4$. The more difficult the integrand, the larger lw should be.
Default: 800$800$
Constraint: lw4${\mathbf{lw}}\ge 4$.
2:     liw – int64int32nag_int scalar
The dimension of the array iw as declared in the (sub)program from which nag_quad_1d_fin_wcauchy (d01aq) is called. The number of sub-intervals into which the interval of integration may be divided cannot exceed liw.
Default: lw / 4${\mathbf{lw}}/4$
Constraint: liw1${\mathbf{liw}}\ge 1$.

None.

### Output Parameters

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

## Error Indicators and Warnings

Note: nag_quad_1d_fin_wcauchy (d01aq) 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 ifail = 1${\mathbf{ifail}}=1$
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 ifail = 2${\mathbf{ifail}}=2$
Round-off error prevents the requested tolerance from being achieved. Consider requesting less accuracy.
W ifail = 3${\mathbf{ifail}}=3$
Extremely bad local behaviour of g(x)$g\left(x\right)$ causes a very strong subdivision around one (or more) points of the interval. The same advice applies as in the case of ${\mathbf{ifail}}={\mathbf{1}}$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, c = a${\mathbf{c}}={\mathbf{a}}$ or c = b${\mathbf{c}}={\mathbf{b}}$.
ifail = 5${\mathbf{ifail}}=5$
 On entry, lw < 4${\mathbf{lw}}<4$, or liw < 1${\mathbf{liw}}<1$.

## Accuracy

nag_quad_1d_fin_wcauchy (d01aq) cannot guarantee, but in practice usually achieves, the following accuracy:
 |I − result| ≤ tol, $|I-result|≤tol,$
where
 tol = max {|epsabs|,|epsrel| × |I|} , $tol=max{|epsabs|,|epsrel|×|I|} ,$
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. $|I-result|≤abserr≤tol.$

The time taken by nag_quad_1d_fin_wcauchy (d01aq) 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_wcauchy (d01aq) along with the integral contributions and error estimates over these sub-intervals.
Specifically, for i = 1,2,,n$i=1,2,\dots ,n$, let ri${r}_{i}$ denote the approximation to the value of the integral over the sub-interval [ai,bi${a}_{i},{b}_{i}$] in the partition of [a,b]$\left[a,b\right]$ and ei${e}_{i}$ be the corresponding absolute error estimate. Then, aibig(x)w(x)dxri$\underset{{a}_{i}}{\overset{{b}_{i}}{\int }}g\left(x\right)w\left(x\right)dx\simeq {r}_{i}$ and result = i = 1nri${\mathbf{result}}=\sum _{i=1}^{n}{r}_{i}$. The value of n$n$ is returned in iw(1)${\mathbf{iw}}\left(1\right)$, and the values ai${a}_{i}$, bi${b}_{i}$, ei${e}_{i}$ and ri${r}_{i}$ are stored consecutively in the array w, that is:
• ai = w(i)${a}_{i}={\mathbf{w}}\left(i\right)$,
• bi = w(n + i)${b}_{i}={\mathbf{w}}\left(n+i\right)$,
• ei = w(2n + i)${e}_{i}={\mathbf{w}}\left(2n+i\right)$ and
• ri = w(3n + i)${r}_{i}={\mathbf{w}}\left(3n+i\right)$.

## Example

```function nag_quad_1d_fin_wcauchy_example
a = -1;
b = 1;
c = 0.5;
epsabs = 0;
epsrel = 0.0001;
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_wcauchy(@g, a, b, c, epsabs, epsrel);
result, abserr, ifail

function result = g(x)
result = 1/(x^2+0.01^2);
```
```

result =

-628.4617

abserr =

0.0132

ifail =

0

```
```function d01aq_example
a = -1;
b = 1;
c = 0.5;
epsabs = 0;
epsrel = 0.0001;
[result, abserr, w, iw, ifail] = d01aq(@g, a, b, c, epsabs, epsrel);
result, abserr, ifail

function result = g(x)
result = 1/(x^2+0.01^2);
```
```

result =

-628.4617

abserr =

0.0132

ifail =

0

```