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Chapter Contents
Chapter Introduction
NAG Toolbox

## Purpose

nag_quad_1d_fin_wsing (d01ap) is an adaptive integrator which calculates an approximation to the integral of a function g(x)w(x)$g\left(x\right)w\left(x\right)$ over a finite interval [a,b]$\left[a,b\right]$:
 b I = ∫ g(x)w(x)dx a
$I= ∫ab g(x) w(x) dx$
where the weight function w$w$ has end point singularities of algebraico-logarithmic type.

## Syntax

[result, abserr, w, iw, ifail] = d01ap(g, a, b, alfa, beta, key, epsabs, epsrel, 'lw', lw, 'liw', liw)
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_wsing(g, a, b, alfa, beta, key, epsabs, epsrel, 'lw', lw, 'liw', liw)

## Description

nag_quad_1d_fin_wsing (d01ap) is based on the QUADPACK routine QAWSE (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)$w\left(x\right)$ may have algebraico-logarithmic singularities at the end points a$a$ and/or b$b$. The strategy is a modification of that in nag_quad_1d_fin_osc (d01ak). We start by bisecting the original interval and applying modified Clenshaw–Curtis integration of orders 12$12$ and 24$24$ to both halves. Clenshaw–Curtis integration is then used on all sub-intervals which have a$a$ or b$b$ as one of their end points (see Piessens et al. (1974)). On the other sub-intervals Gauss–Kronrod (7$7$15$15$ point) integration is carried out.
A ‘global’ acceptance criterion (as defined by Malcolm and Simpson (1976)) is used. The local error estimation control is described in 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, Mertens I and Branders M (1974) Integration of functions having end-point singularities Angew. Inf. 16 65–68

## 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.
Constraint: b > a${\mathbf{b}}>{\mathbf{a}}$.
4:     alfa – double scalar
The parameter α$\alpha$ in the weight function.
Constraint: alfa > 1.0${\mathbf{alfa}}>-1.0$.
5:     beta – double scalar
The parameter β$\beta$ in the weight function.
Constraint: beta > 1.0${\mathbf{beta}}>-1.0$.
6:     key – int64int32nag_int scalar
Indicates which weight function is to be used.
key = 1${\mathbf{key}}=1$
w(x) = (xa)α(bx)β$w\left(x\right)={\left(x-a\right)}^{\alpha }{\left(b-x\right)}^{\beta }$.
key = 2${\mathbf{key}}=2$
w(x) = (xa)α(bx)βln(xa)$w\left(x\right)={\left(x-a\right)}^{\alpha }{\left(b-x\right)}^{\beta }\mathrm{ln}\left(x-a\right)$.
key = 3${\mathbf{key}}=3$
w(x) = (xa)α(bx)βln(bx)$w\left(x\right)={\left(x-a\right)}^{\alpha }{\left(b-x\right)}^{\beta }\mathrm{ln}\left(b-x\right)$.
key = 4${\mathbf{key}}=4$
w(x) = (xa)α(bx)βln(xa)ln(bx)$w\left(x\right)={\left(x-a\right)}^{\alpha }{\left(b-x\right)}^{\beta }\mathrm{ln}\left(x-a\right)\mathrm{ln}\left(b-x\right)$.
Constraint: key = 1${\mathbf{key}}=1$, 2$2$, 3$3$ or 4$4$.
7:     epsabs – double scalar
The absolute accuracy required. If epsabs is negative, the absolute value is used. See Section [Accuracy].
8:     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_wsing (d01ap) 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: lw8${\mathbf{lw}}\ge 8$.
2:     liw – int64int32nag_int scalar
The dimension of the array iw as declared in the (sub)program from which nag_quad_1d_fin_wsing (d01ap) 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: liw2${\mathbf{liw}}\ge 2$.

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_wsing (d01ap) 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 the position of a discontinuity or a singularity of algebraico-logarithmic type within the interval can be determined, the interval must be split up at this point and the integrator called 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.
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 integrand behaviour causes a very strong subdivision around one (or more) points of the interval. The same advice applies as in the case of ${\mathbf{ifail}}={\mathbf{1}}$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, b ≤ a${\mathbf{b}}\le {\mathbf{a}}$, or alfa ≤ − 1.0${\mathbf{alfa}}\le -1.0$, or beta ≤ − 1.0${\mathbf{beta}}\le -1.0$, or key ≠ 1${\mathbf{key}}\ne 1$, 2$2$, 3$3$ or 4$4$.
ifail = 5${\mathbf{ifail}}=5$
 On entry, lw < 8${\mathbf{lw}}<8$, or liw < 2${\mathbf{liw}}<2$.

## Accuracy

nag_quad_1d_fin_wsing (d01ap) 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_wsing (d01ap) 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_wsing (d01ap) 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] $\left[{a}_{i},{b}_{i}\right]$ in the partition of [a,b] $\left[a,b\right]$ and ei ${e}_{i}$ be the corresponding absolute error estimate. Then, aibi f(x) w(x) dx ri $\underset{{a}_{i}}{\overset{{b}_{i}}{\int }}f\left(x\right)w\left(x\right)dx\simeq {r}_{i}$ and result = i = 1n ri ${\mathbf{result}}=\sum _{i=1}^{n}{r}_{i}$. 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_wsing_example
a = 0;
b = 1;
alfa = 0;
beta = 0;
key = int64(2);
epsabs = 0;
epsrel = 0.0001;
g = @(x) cos(10*pi*x);
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_wsing(g, a, b, alfa, beta, key, epsabs, epsrel);
result, abserr, ifail
```
```

result =

-0.0490

abserr =

1.1390e-07

ifail =

0

```
```function d01ap_example
a = 0;
b = 1;
alfa = 0;
beta = 0;
key = int64(2);
epsabs = 0;
epsrel = 0.0001;
g = @(x) cos(10*pi*x);
[result, abserr, w, iw, ifail] = d01ap(g, a, b, alfa, beta, key, epsabs, epsrel);
result, abserr, ifail
```
```

result =

-0.0490

abserr =

1.1390e-07

ifail =

0

```