# Documentation

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## 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. It accepts a vector x and returns a vector y, the function fun evaluated at each element of x. Limits a and b must be finite.

Parameterizing Functions 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 might be more efficient with low accuracies or nonsmooth integrands.

• The quad function might be most efficient for low accuracies with nonsmooth integrands.

• The quadl function might be more efficient than quad at higher accuracies with smooth integrands.

• The quadgk function might be most efficient for high accuracies and oscillatory integrands. It supports infinite intervals and can handle moderate singularities at the endpoints. It also supports contour integration along piecewise linear paths.

• If the interval is infinite, $\left[a,\infty \right)$, then for the integral of fun(x) to exist, fun(x) must decay as x approaches infinity, and quadgk requires it to decay rapidly. Special methods should be used for oscillatory functions on infinite intervals, but quadgk can be used if fun(x) decays fast enough.

• The quadgk function will integrate functions that are singular at finite endpoints if the singularities are not too strong. For example, it will integrate functions that behave at an endpoint c like log|x-c| or |x-c|p for p >= -1/2. If the function is singular at points inside (a,b), write the integral as a sum of integrals over subintervals with the singular points as endpoints, compute them with quadgk, and add the results.

## Examples

Pass the function handle, @myfun, to quadl:

where the function 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);

## Diagnostics

quadl might 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.

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## 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.