quadl (MATLAB Functions)

Numerically evaluate integral, adaptive Lobatto quadrature

Syntax

q = quadl(fun,a,b)
q = quadl(fun,a,b,tol)
q = quadl(fun,a,b,tol,trace)
[q,fcnt] = quadl(fun,a,b,...)

Description

q = quadl(fun,a,b) approximates the integral of function fun from a to b, to within an error of 10^-6 using recursive adaptive Lobatto quadrature. fun is a function handle. See Function Handles in the MATLAB Programming documentation for more information. fun accepts a vector x and returns a vector y, the function fun evaluated at each element of x.

Parameterizing Functions Called by Function Functions, in the MATLAB mathematics documentation, explains how to provide additional parameters to the function fun, if necessary.

q = quadl(fun,a,b,tol) uses an absolute error tolerance of tol instead of the default, which is 1.0e-6. Larger values of tol result in fewer function evaluations and faster computation, but less accurate results.

quadl(fun,a,b,tol,trace) with non-zero trace shows the values of [fcnt a b-a q] during the recursion.

[q,fcnt] = quadl(...) returns the number of function evaluations.

Use array operators .*, ./ and .^ in the definition of fun so that it can be evaluated with a vector argument.

The function quad may be more efficient with low accuracies or nonsmooth integrands.

Examples

Pass M-file function handle @myfun to quadl:

```
Q = quadl(@myfun,0,2);
```

where the M-file myfun.m is

function y = myfun(x) 
y = 1./(x.^3-2*x-5);

Pass anonymous function handle F to quadl:

F = @(x) 1./(x.^3-2*x-5);
Q = quadl(F,0,2);

Algorithm

quadl implements a high order method using an adaptive Gauss/Lobatto quadrature rule.

Diagnostics

quadl may issue one of the following warnings:

'Minimum step size reached' indicates that the recursive interval subdivision has produced a subinterval whose length is on the order of roundoff error in the length of the original interval. A nonintegrable singularity is possible.

'Maximum function count exceeded' indicates that the integrand has been evaluated more than 10,000 times. A nonintegrable singularity is likely.

'Infinite or Not-a-Number function value encountered' indicates a floating point overflow or division by zero during the evaluation of the integrand in the interior of the interval.

References

[1] Gander, W. and W. Gautschi, "Adaptive Quadrature - Revisited", BIT, Vol. 40, 2000, pp. 84-101. This document is also available at http:// www.inf.ethz.ch/personal/gander.

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