# Numerical integration: quadgk vs integral

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Mariano on 6 Mar 2015
Answered: Mariano on 3 Jul 2019
Hello: I have not been able to find out what is the underlying quadrature formula in Matlab's builtin function integral. I think that quad uses Simpson, quadl a Gauss-Lobato formula and quadgk the Gauss-Kronrod formula G7K15, and all of them use some kind of adaptative scheme. What does integral(f,a,b) do with my poor function f? When should I use integral and when quadgk? Thank you, Mariano

Christiaan on 9 Mar 2015
Edited: Christiaan on 9 Mar 2015
Dear Mariano,
The Mathlab function 'integral(fun,xmin,xmax)' numerically integrates function fun from xmin to xmax using global adaptive quadrature and default error tolerances.
In the documention of this function (doc integral), this ref can be found:
% Portions based on "quadva" by Lawrence F. Shampine.
% Journal of Computational and Applied Mathematics 211, 2008, pp.131-140
In the Documentation of quadgk you can find a list of multiple quadrature functions that can be used whith there (dis)advantages.
Kind regards, Christiaan
TheStranger on 3 Sep 2017
Strange, I tried both of them on some pretty complicated function and the answers differ by ~10^(-8) Also, for some strange reason, integral does not have an option "MaxIntervalCount", while quadgk does.
Josh Meyer on 2 Jul 2019
To summarize the differences between the two functions:
• integral and quadgk use the same quadrature method as described in the reference paper
• integral and quadgk use different default error tolerances for RelTol and AbsTol
• quadgk has the MaxIntervalCount option, and uses a relatively small default value compared to integral. This means that with integral you will rarely if ever need to adjust this parameter (hence why it isn't there). This would only be with highly oscillatory integrands that are difficult to evaluate.
• quadgk has an extra output that approxmates the absolute error. This lets you specify MaxIntervalCount, check the error in the calculation, and adjust MaxIntervalCount as needed.

Mariano on 3 Jul 2019
Best wishes,
Mariano

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