(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.

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

`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 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, $$\left[a,\infty \right)$$, 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|`

for^{p}`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

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

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); 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://www.inf.ethz.ch/personal/gander`

.

`dblquad`

| `integral`

| `integral2`

| `integral3`

| `quad2d`

| `quadgk`

| `quadl`

| `quadv`

| `trapz`

| `triplequad`