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Chapter Contents
Chapter Introduction
NAG Toolbox

## Purpose

nag_quad_1d_indef (d01ar) computes definite and indefinite integrals over a finite range to a specified relative or absolute accuracy, using the method described in Patterson (1968).

## Syntax

[acc, ans, n, alpha, ifail] = d01ar(a, b, fun, relacc, absacc, maxrul, iparm, alpha)
[acc, ans, n, alpha, ifail] = nag_quad_1d_indef(a, b, fun, relacc, absacc, maxrul, iparm, alpha)

## Description

nag_quad_1d_indef (d01ar) evaluates definite and indefinite integrals of the form:
 b ∫ f(t)dt a
$∫abf(t)dt$
using the method described in Patterson (1968).

### Definite Integrals

In this case nag_quad_1d_indef (d01ar) must be called with iparm = 0${\mathbf{iparm}}=0$. By linear transformation the integral is changed to
 + 1 I = ∫ F(x)dx − 1
$I=∫-1 +1F(x)dx$
where
 F(x) = (b − a)/2 f ((b + a + (b − a)x)/2) $F(x)=b-a2 f (b+a+(b-a)x2)$
and is then approximated by an n$n$-point quadrature rule
 n I = ∑ wkF(xk) k = 1
$I=∑k=1nwkF(xk)$
where wk${w}_{k}$ are the weights and xk${x}_{k}$ are the abscissae.
The function uses a family of nine interlacing rules based on the optimal extension of the three-point Gauss rule. These rules use 1$1$, 3$3$, 7$7$, 15$15$, 31$31$, 63$63$, 127$127$, 255$255$ and 511$511$ points and have respective polynomial integrating degrees 1$1$, 5$5$, 11$11$, 23$23$, 47$47$, 95$95$, 191$191$, 383$383$ and 767$767$. Each rule has the property that the next in sequence includes all the points of its predecessor and has the greatest possible increase in integrating degree.
The integration method is based on the successive application of these rules until the absolute value of the difference of two successive results differs by not more than absacc, or relatively by not more than relacc. The result of the last rule used is taken as the value of the integral (ans), and the absolute difference of the results of the last two rules used is taken as an estimate of the absolute error (acc). Due to their interlacing form no integrand evaluations are wasted in passing from one rule to the next.

### Indefinite Integrals

Suppose the value of the integral
 d ∫ f(t)dt c
$∫cdf(t)dt$
is required for a number of sub-intervals [c,d]$\left[c,d\right]$, all of which lie in an interval [a,b]$\left[a,b\right]$.
In this case nag_quad_1d_indef (d01ar) should first be called with the parameter iparm = 1${\mathbf{iparm}}=1$ and the interval set to [a,b]$\left[a,b\right]$. The function then calculates the integral over [a,b]$\left[a,b\right]$ and the Legendre expansion of the integrand, using the same integrand values. If the function is subsequently called with iparm = 2${\mathbf{iparm}}=2$ and the interval set to [c,d]$\left[c,d\right]$, the integral over [c,d]$\left[c,d\right]$ is calculated by analytical integration of the Legendre expansion, without further evaluations of the integrand.
For the interval [1,1]$\left[-1,1\right]$ the expansion takes the form
 ∞ F(x) = ∑ αiPi(x) i = 0
$F(x)=∑i=0∞αiPi(x)$
where Pi(x)${P}_{i}\left(x\right)$ is the order i$i$ Legendre polynomial. Assuming that the integral over the full range [1,1]$\left[-1,1\right]$ was evaluated to the required accuracy using an n$n$-point rule, then the coefficients
 + 1 αi = (1/2)(2i − 1) ∫ Pi(x)F(x)dx,  i = 0,1, … ,m − 1
$αi=12(2i-1)∫-1 +1Pi(x)F(x)dx, i=0,1,…,m$
are evaluated by that same rule, up to
 m = (3n − 1) / 4. $m=(3n- 1)/4.$
The accuracy for indefinite integration should be of the same order as that obtained for the definite integral over the full range. The indefinite integrals will be exact when F(x)$F\left(x\right)$ is a polynomial of degree m$\text{}\le m$.

## References

Patterson T N L (1968) The Optimum addition of points to quadrature formulae Math. Comput. 22 847–856

## Parameters

### Compulsory Input Parameters

1:     a – double scalar
a$a$, the lower limit of integration.
2:     b – double scalar
b$b$, the upper limit of integration. It is not necessary that a < b$a.
3:     fun – function handle or string containing name of m-file
fun must return the value of the integrand f$f$ at a specified point.
[result] = fun(x)

Input Parameters

1:     x – double scalar
The point in [a,b]$\left[a,b\right]$ at which the integrand f$f$ must be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.
If iparm = 2${\mathbf{iparm}}=2$, fun is not called.
4:     relacc – double scalar
The relative accuracy required. If convergence according to absolute accuracy is required, relacc should be set to zero (but see also Section [Accuracy]). If relacc < 0.0${\mathbf{relacc}}<0.0$, its absolute value is used.
If iparm = 2${\mathbf{iparm}}=2$, relacc is not used.
5:     absacc – double scalar
The absolute accuracy required. If convergence according to relative accuracy is required, absacc should be set to zero (but see also Section [Accuracy]). If absacc < 0.0${\mathbf{absacc}}<0.0$, its absolute value is used.
If iparm = 2${\mathbf{iparm}}=2$, absacc is not used.
6:     maxrul – int64int32nag_int scalar
The maximum number of successive rules that may be used.
Constraint: 1maxrul9$1\le {\mathbf{maxrul}}\le 9$. If maxrul is outside these limits, the value 9$9$ is assumed.
If iparm = 2${\mathbf{iparm}}=2$, maxrul is not used.
7:     iparm – int64int32nag_int scalar
Indicates the task to be performed by the function.
iparm = 0${\mathbf{iparm}}=0$
Only the definite integral over [a,b]$\left[a,b\right]$ is evaluated.
iparm = 1${\mathbf{iparm}}=1$
As well as the definite integral, the expansion of the integrand in Legendre polynomials over [a,b]$\left[a,b\right]$ is calculated, using the same values of the integrand as used to compute the integral. The expansion coefficients, and some other quantities, are returned in alpha for later use in computing indefinite integrals.
iparm = 2${\mathbf{iparm}}=2$
f(t)$f\left(t\right)$ is integrated analytically over [a,b]$\left[a,b\right]$ using the previously computed expansion, stored in alpha. No further evaluations of the integrand are required. The function must previously have been called with iparm = 1${\mathbf{iparm}}=1$ and the interval [a,b]$\left[a,b\right]$ must lie within that specified for the previous call. In this case only the arguments a, b, iparm, ans, alpha and ifail are used.
Constraint: iparm = 0${\mathbf{iparm}}=0$, 1$1$ or 2$2$.
8:     alpha(390$390$) – double array
If iparm = 2${\mathbf{iparm}}=2$, alpha must contain the coefficients of the Legendre expansions of the integrand, as returned by a previous call of nag_quad_1d_indef (d01ar) with iparm = 1${\mathbf{iparm}}=1$ and a range containing the present range.
If iparm = 0${\mathbf{iparm}}=0$ or 1$1$, alpha need not be set on entry.

None.

None.

### Output Parameters

1:     acc – double scalar
If iparm = 0${\mathbf{iparm}}=0$ or 1$1$, acc contains the absolute value of the difference between the last two successive estimates of the integral. This may be used as a measure of the accuracy actually achieved.
If iparm = 2${\mathbf{iparm}}=2$, acc is not used.
2:     ans – double scalar
The estimated value of the integral.
3:     n – int64int32nag_int scalar
When iparm = 0${\mathbf{iparm}}=0$ or 1$1$, n contains the number of integrand evaluations used in the calculation of the integral.
If iparm = 2${\mathbf{iparm}}=2$, n is not used.
4:     alpha(390$390$) – double array
If iparm = 1${\mathbf{iparm}}=1$, the first m$m$ elements of alpha hold the coefficients of the Legendre expansion of the integrand, and the value of m$m$ is stored in alpha(390)${\mathbf{alpha}}\left(390\right)$. alpha must not be changed between a call with iparm = 1${\mathbf{iparm}}=1$ and subsequent calls with iparm = 2${\mathbf{iparm}}=2$.
If iparm = 2${\mathbf{iparm}}=2$, the first m$m$ elements of alpha are unchanged on exit.
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_indef (d01ar) 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${\mathbf{ifail}}=1$
If iparm = 0${\mathbf{iparm}}=0$ or 1$1$, this indicates that all maxrul rules have been used and the integral has not converged to the accuracy requested. In this case ans contains the last approximation to the integral, and acc contains the difference between the last two approximations. To check this estimate of the integral, nag_quad_1d_indef (d01ar) could be called again to evaluate
 b c b ∫ f(t)dt  as ∫ f(t)dt + ∫ f(t)dt  for some ​a < c < b. a a c
$∫abf(t)dt as ∫ac f(t)dt+∫cb f(t)dt for some ​a
If iparm = 2${\mathbf{iparm}}=2$, this indicates failure of convergence during the run with iparm = 1${\mathbf{iparm}}=1$ in which the Legendre expansion was created.
ifail = 2${\mathbf{ifail}}=2$
On entry, iparm0${\mathbf{iparm}}\ne 0$, 1$1$ or 2$2$
ifail = 3${\mathbf{ifail}}=3$
The function is called with iparm = 2${\mathbf{iparm}}=2$ but a previous call with iparm = 1${\mathbf{iparm}}=1$ has been omitted or was invoked with an integration interval of length zero.
ifail = 4${\mathbf{ifail}}=4$
On entry, with iparm = 2${\mathbf{iparm}}=2$, the interval for indefinite integration is not contained within the interval specified when nag_quad_1d_indef (d01ar) was previously called with iparm = 1${\mathbf{iparm}}=1$.

## Accuracy

The relative or absolute accuracy required is specified by you in the variables relacc or absacc. nag_quad_1d_indef (d01ar) will terminate whenever either the relative accuracy specified by relacc or the absolute accuracy specified by absacc is reached. One or other of these criteria may be ‘forced’ by setting the parameter for the other to zero. If both relacc and absacc are specified as zero, then the function uses the value 10.0 × (machine precision) for relacc.
If on exit ${\mathbf{ifail}}={\mathbf{0}}$, then it is likely that the result is correct to one or other of these accuracies. If on exit ${\mathbf{ifail}}={\mathbf{1}}$, then it is likely that neither of the requested accuracies has been reached.
When you have no prior idea of the magnitude of the integral, it is possible that an unreasonable accuracy may be requested, e.g., a relative accuracy for an integral which turns out to be zero, or a small absolute accuracy for an integral which turns out to be very large. Even if failure is reported in such a case, the value of the integral may still be satisfactory. The device of setting the other ‘unused’ accuracy parameter to a small positive value (e.g., 109${10}^{-9}$ for an implementation of 11$11$-digit precision) rather than zero, may prevent excessive calculation in such a situation.
To avoid spurious convergence, it is recommended that relative accuracies larger than about 103${10}^{-3}$ be avoided.

The time taken by nag_quad_1d_indef (d01ar) depends on the complexity of the integrand and the accuracy required.
This function uses the Patterson method over the whole integration interval and should therefore be suitable for well behaved functions. However, for very irregular functions it would be more efficient to submit the differently behaved regions separately for integration.

## Example

```function nag_quad_1d_indef_example
a = 0;
b = 1;
fun = @(x) 4/(1+x^2);
relacc = 0;
absacc = 1e-05;
maxrul = int64(0);
iparm = int64(0);
alpha = zeros(390,1);
[acc, ans, n, alphaOut, ifail] = ...
nag_quad_1d_indef(a, b, fun, relacc, absacc, maxrul, iparm, alpha);
acc, ans, n, ifail
```
```

acc =

1.8377e-08

ans =

3.1416

n =

15

ifail =

0

```
```function d01ar_example
a = 0;
b = 1;
fun = @(x) 4/(1+x^2);
relacc = 0;
absacc = 1e-05;
maxrul = int64(0);
iparm = int64(0);
alpha = zeros(390,1);
[acc, ans, n, alphaOut, ifail] = ...
d01ar(a, b, fun, relacc, absacc, maxrul, iparm, alpha);
acc, ans, n, ifail
```
```

acc =

1.8377e-08

ans =

3.1416

n =

15

ifail =

0

```