Numerically evaluate integral, adaptive GaussKronrod quadrature
q = quadgk(fun,a,b)
[q,errbnd] = quadgk(fun,a,b)
[q,errbnd] = quadgk(fun,a,b,param1,val1,param2,val2,...)
q = quadgk(fun,a,b)
attempts
to approximate the integral of a scalarvalued function fun
from a
to b
using
highorder global adaptive quadrature and default error tolerances.
The function y = fun(x)
should accept a vector
argument x
and return a vector result y
,
where y
is the integrand evaluated at each element
of x
. fun
must be a function
handle. Limits a
and b
can be Inf
or Inf
.
If both are finite, they can be complex. If at least one is complex,
the integral is approximated over a straight line path from a
to b
in
the complex plane.
Parameterizing Functions explains how to provide additional
parameters to the function fun
, if necessary.
[q,errbnd] = quadgk(fun,a,b)
returns
an approximate upper bound on the absolute error, Q  I
,
where I
denotes the exact value of the integral.
[q,errbnd] = quadgk(fun,a,b,param1,val1,param2,val2,...)
performs
the integration with specified values of optional parameters. The
available parameters are
Parameter  Description  

 Absolute error tolerance. The default value
of 

 Relative error tolerance. The default value
of  
 Vector of integration waypoints.  If If 
 Maximum number of intervals allowed. The
default value is  The 
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 arrayvalued fun
.
If the interval is infinite, [a,Inf)
,
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 logxc
or xc^{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.
Integrand with a singularity at an integration end point
Write a function myfun
that computes the
integrand:
function y = myfun(x) y = exp(x).*log(x);
Then pass @myfun
, a function handle to myfun
,
to quadgk
, along with the limits of integration, 0
to 1
:
q = quadgk(@myfun,0,1) q = 1.3179
Alternatively, you can pass the integrand to quadgk
as
an anonymous function handle F
:
f = (@(x)exp(x).*log(x)); q = quadgk(f,0,1);
Oscillatory integrand on a semiinfinite interval
Integrate over a semiinfinite interval with specified tolerances, and return the approximate error bound:
f = @(x)x.^5.*exp(x).*sin(x); [q,errbnd] = quadgk(f,0,inf,'RelTol',1e8,'AbsTol',1e12) q = 15.0000 errbnd = 9.4386e009
Contour integration around a pole
Use Waypoints
to integrate around a pole
using a piecewise linear contour:
f = @(z)1./(2*z  1); q = quadgk(f,1i,1i,'Waypoints',[1i,1+i,1+i]) q = 0.0000 + 3.1416i
quadgk
may issue one of the following warnings:
'Minimum step size reached'
indicates that
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.
'Reached the limit on the maximum number of intervals
in use'
indicates that the integration was terminated before
meeting the tolerance requirements and that continuing the integration
would require more than MaxIntervalCount
subintervals.
The integral may not exist, or it may be difficult to approximate
numerically. Increasing MaxIntervalCount
usually
does not help unless the tolerance requirements were nearly met when
the integration was previously terminated.
'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.
[1] L.F. Shampine "Vectorized Adaptive Quadrature in MATLAB," Journal of Computational and Applied Mathematics, 211, 2008, pp.131–140.
dblquad
 function_handle
 integral
 integral2
 integral3
 quad
 quad2d
 quadl
 quadv
 triplequad