`quad` is not recommended. Use `integral` instead.

## Syntax

``q = quad(fun,a,b)``
``q = quad(fun,a,b,tol)``
``q = quad(fun,a,b,tol,trace)``
``[q,fcnEvals] = quad(___)``

## Description

example

````q = quad(fun,a,b)` approximates the integral of function `fun` from `a` to `b` using recursive adaptive Simpson quadrature:$q=\underset{a}{\overset{b}{\int }}f\left(x\right)dx$```
````q = quad(fun,a,b,tol)` specifies an absolute error tolerance `tol` for each subinterval, instead of the default value of `1e-6`.```
````q = quad(fun,a,b,tol,trace)` optionally turns on the display of diagnostic information. When `trace` is nonzero, `quad` shows the vector of values ```[fcnEvals, a, b-a, Q]``` during the recursion.```
````[q,fcnEvals] = quad(___)` additionally returns the number of function evaluations `fcnEvals`. You can specify any of the previous input argument combinations.```

## Examples

collapse all

Compute the integral

`${\int }_{0}^{2}\frac{1}{{x}^{3}-2x-5}dx.$`

First, create an anonymous function `myfun` that computes the integrand.

`myfun = @(x) 1./(x.^3-2*x-5);`

Now use `quad` to compute the integral. Specify the limits of integration as the second and third input arguments.

`q = quad(myfun,0,2)`
```q = -0.4605 ```

Alternatively, you can pass the integrand to `quad` by creating a function file:

```function y = myfun(x) y = 1./(x.^3-2*x-5); end ```

With this method, the call to `quad` becomes `quad(@myfun,0,2)`.

## Input Arguments

collapse all

Integrand, specified as a function handle that defines the function to be integrated from `a` to `b`.

For scalar-valued problems, the function `y = fun(x)` must accept a vector argument `x` and return a vector result `y`, where `y` is the integrand evaluated at each element of `x`. This requirement generally means that `fun` must use array operators (`.^`, `.*`, …) instead of matrix operators (`^`, `*`, …).

Parameterizing Functions explains how to provide additional parameters to the function `fun`, if necessary.

Example: `q = quad(@(x) exp(1-x.^2),a,b)` integrates an anonymous function handle.

Example: `q = quad(@myFun,a,b)` integrates the function `myFun`, which is saved as a file.

Data Types: `function_handle`

Integration limits, specified as separate scalar arguments. The limits `a` and `b` must be finite.

Example: `quad(fun,0,1)` integrates `fun` from `0` to `1`.

Data Types: `single` | `double`

Absolute error tolerance, specified as a scalar. `quad` uses the absolute error tolerance on each subinterval in the integration. As the magnitude of `tol` increases, `quad` performs fewer function evaluations and completes the calculation faster, but produces less accurate results.

Example: `quad(fun,a,b,1e-12)` sets the absolute error tolerance to `1e-12`.

Data Types: `single` | `double`

Toggle for diagnostic information, specified as a nonzero scalar. When `trace` is nonzero, `quad` displays the vector of values `[fcnEvals, a, b-a, Q]` for each subinterval in the recursion:

• `fcnEvals` gives the number of function evaluations

• `a` and `b` are the limits of integration

• `Q` is the computed area of the subinterval

Example: `quad(fun,a,b,1e-8,1)` integrates `fun` from `a` to `b` with a tolerance of `1e-8` and diagnostic information turned on.

## Output Arguments

collapse all

Value of integral, returned as a scalar.

Number of function evaluations, returned as a scalar.

## Algorithms

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

## References

[1] Gander, W., and W. Gautschi. “Adaptive Quadrature—Revisited.” BIT Numerical Mathematics 40 (2000): 84–101. https://doi.org/10.1023/A:1022318402393