MATLAB Function Reference 
Numerically evaluate integral, adaptive Simpson quadrature
Note
The quad8 function, which implemented a higher order method, is obsolete. The quadl function is its recommended replacement.

Syntax
q = quad(fun,a,b) q = quad(fun,a,b,tol) q = quad(fun,a,b,tol,trace) q = quad(fun,a,b,tol,trace,p1,p2,...) [q,fcnt] = quadl(fun,a,b,...)
Description
Quadrature is a numerical method used to find the area under the graph of a function, that is, to compute a definite integral.
q = quad(fun,a,b)
approximates the integral of function fun
from a
to b
to within an error of 10^{6} using recursive adaptive Simpson quadrature. fun
accepts a vector x
and returns a vector y
, the function fun
evaluated at each element of x
.
q = quad(fun,a,b,tol)
uses an absolute error tolerance tol instead of the default which is 1.0e6
. Larger values of tol
result in fewer function evaluations and faster computation, but less accurate results. In MATLAB version 5.3 and earlier, the quad
function used a less reliable algorithm and a default relative tolerance of 1.0e3
.
q = quad(fun,a,b,tol,trace)
with nonzero trace
shows the values of [fcnt a ba Q]
during the recursion.
provides for additional arguments q = quad(
fun,a,b,
tol,
trace,p1,p2,...)
p1,p2,...
to be passed directly to function fun
, fun(x,p1,p2,...)
. Pass empty matrices for tol
or trace
to use the default values.
[q,fcnt] = quad(...)
returns the number of function evaluations.
The function quadl
may be more efficient with high accuracies and smooth integrands.
Examples
Algorithm
quad
implements a low order method using an adaptive recursive Simpson's rule.
Diagnostics
quad
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 NotaNumber function value encountered'
indicates a floating point overflow or division by zero during the evaluation of the integrand in the interior of the interval.
See Also
dblquad
, inline
, quadl
, triplequad
, @
(function handle)
References
[1] Gander, W. and W. Gautschi, "Adaptive Quadrature  Revisited", BIT, Vol. 40, 2000, pp. 84101. This document is also available at http:// www.inf.ethz.ch/personal/gander.
qrupdate  quadl 