# Documentation

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quadv will be removed in a future release. Use integral with the 'ArrayValued' option instead.

## Description

Q = quadv(fun,a,b) approximates the integral of the complex array-valued function fun from a to b to within an error of 1.e-6 using recursive adaptive Simpson quadrature. fun is a function handle. The function Y = fun(x) should accept a scalar argument x and return an array result Y, whose components are the integrands evaluated at x. Limits a and b must be finite.

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

Q = quadv(fun,a,b,tol) uses the absolute error tolerance tol for all the integrals instead of the default, which is 1.e-6.

 Note   The same tolerance is used for all components, so the results obtained with quadv are usually not the same as those obtained with quad on the individual components.

Q = quadv(fun,a,b,tol,trace) with non-zero trace shows the values of [fcnt a b-a Q(1)] during the recursion.

[Q,fcnt] = quadv(...) returns the number of function evaluations.

• The quad function might be most efficient for low accuracies with nonsmooth integrands.

• The quadl function might be more efficient than quad at higher accuracies with smooth integrands.

• The quadgk function might 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, $\left[a,\infty \right)$, 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|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

For the parameterized array-valued function myarrayfun, defined by

function Y = myarrayfun(x,n)
Y = 1./((1:n)+x);

the following command integrates myarrayfun, for the parameter value n = 10 between a = 0 and b = 1:

The resulting array Qv has 10 elements estimating Q(k) = log((k+1)./(k)), for k = 1:10.

The entries in Qv are slightly different than if you compute the integrals using quad in a loop:

for k = 1:10
end

where myscalarfun is:

function y = myscalarfun(x,k)
y = 1./(k+x);