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# Documentation

## Description

Quadrature is a numerical method used to find the area under the graph of a function, that is, to compute a definite integral.

$q=\underset{a}{\overset{b}{\int }}f\left(x\right)dx$

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 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.

• 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 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.

## Examples

To compute the integral

$\underset{0}{\overset{2}{\int }}\frac{1}{{x}^{3}-2x-5}dx,$

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);

## 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 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.

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### Algorithms

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

## References

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