MATLAB Function Reference    
quad, quad8

Numerically evaluate integral, adaptive Simpson quadrature



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 1.0e-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 = quad(fun,a,b,tol,trace,p1,p2,...) provides for additional arguments 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.


The function fun can be


quad implements a low order method using an adaptive recursive Simpson's rule.


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.

See Also

dblquad, inline, quadl, triplequad, @ (function handle)


[1]  Gander, W. and W. Gautschi, "Adaptive Quadrature - Revisited", BIT, Vol. 40, 2000, pp. 84-101. This document is also available at http://

  qrupdate quadl