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(Not recommended) Numerically evaluate integral, adaptive Lobatto quadrature

`quadl`

is not recommended. Use `integral`

instead.

`q = quadl(fun,a,b)`

q = quadl(fun,a,b,tol)

quadl(fun,a,b,tol,trace)

[q,fcnt] = quadl(...)

`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 list below contains information to help you determine which quadrature function in
MATLAB^{®} to use:

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.The

`quadv`

function vectorizes`quad`

for an array-valued`fun`

.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|`

for^{p}`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.

Pass the function handle, `@myfun`

, to
`quadl`

:

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

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); Q = quadl(F,0,2);

`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.

`quadl`

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

[1] Gander, W. and W. Gautschi, “Adaptive Quadrature
– Revisited,” BIT, Vol. 40, 2000, pp. 84-101. This document is also
available at `https://people.inf.ethz.ch/gander/`

.