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Quadrature of the absolute value of a function

version 1.1 (26 KB) by Jan Valdman
Quadrature of the absolute value of a function

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Updated 19 Mar 2016

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Given a set x of discrete points in 1D and a set of values y=f(x) at these points, a Matlab function trapz(x,abs(y)) provides a quadrature based on a piece-wise linear interpolation of a function f (known as the trapezoidal rule).
Our new function trapzAbs(x,y) detects intervals (x_i, x_{i+1}), where y(x_i)*y(x_{i+1))>0 and applies the regula falsi method and integrates the absolute value of the linear interpolant exactly. Consequently, the approximation of the solution f(x)=0 is provided.
In 2D, this idea it generalized for a P1 approximation (a piecewise affine and globally continuous function know the the finite element method), defined of given triangulation. Then, triangles with vertices v1=(x1,y1), v2=(x2,y2), v3=(x3,y3) satisfying f(v1)*f(v2)*f(v3) >0 are detected and the integral of the absolute value of the function is computed exactly. Similarly, the approximation of the solution f(x,y)=0 is provided.

Our focus is to integrate the absolute value (or its branches) of a given P1 approximation in 1D and 2D exactly.

To test the code, run

example_1D or comparison_1D or example_2D or comparison_2D.

The ideas will be explained in the forthcoming Bc. thesis of Jiri Kadlec written under the supervision of Jan Valdman at the University of South Bohemia in Ceske Budejovice.

Cite As

Jan Valdman (2021). Quadrature of the absolute value of a function (https://www.mathworks.com/matlabcentral/fileexchange/54183-quadrature-of-the-absolute-value-of-a-function), MATLAB Central File Exchange. Retrieved .

MATLAB Release Compatibility
Created with R2014b
Compatible with any release
Platform Compatibility
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