## Quadrature of the absolute value of a function

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Quadrature of the absolute value of a function

Updated Sat, 19 Mar 2016 20:37:40 +0000

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 (2022). 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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