(Not recommended) Numerically evaluate integral, adaptive Simpson quadrature
quad
is not recommended. Use integral
instead.
q = quad(fun,a,b)
q = quad(fun,a,b,tol)
q = quad(fun,a,b,tol,trace)
[q,fcnt] = quad(...)
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)
tries to approximate the
integral of function fun
from a
to
b
to within an error of 1e-6
using recursive
adaptive Simpson quadrature. fun
is a function handle. Limits
a
and b
must be finite. The function y =
fun(x)
should accept a vector argument x
and return a
vector result y
, the integrand evaluated at each element of
x
.
Parameterizing Functions explains how to
provide additional parameters to the function fun
, if necessary.
q = quad(fun,a,b,tol)
uses an absolute error
tolerance 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. In MATLAB® version 5.3 and earlier, the quad
function used a less
reliable algorithm and a default relative tolerance of 1.0e-3
.
q = quad(fun,a,b,tol,trace)
with non-zero
trace
shows the values of
[fcnt a b-a Q]
during the recursion.
[q,fcnt] = quad(...)
returns the number of
function evaluations.
The function quadl
may be more efficient with high accuracies and
smooth integrands.
The list below contains information to help you determine which quadrature function in MATLAB to use:
The quad
function may be most efficient for low accuracies
with nonsmooth integrands.
The quadl
function may be more efficient than
quad
at higher accuracies with smooth integrands.
The quadgk
function may 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, , 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.
To compute the integral
write a function myfun
that computes the integrand:
function y = myfun(x) y = 1./(x.^3-2*x-5);
Then pass @myfun
, a function handle to myfun
, to
quad
, along with the limits of integration, 0
to
2
:
Q = quad(@myfun,0,2) Q = -0.4605
Alternatively, you can pass the integrand to quad
as an anonymous
function handle F
:
F = @(x)1./(x.^3-2*x-5); Q = quad(F,0,2);
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 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.
[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/
.
dblquad
| integral
| integral2
| integral3
| quad2d
| quadgk
| quadl
| quadv
| trapz
| triplequad