Quadrature of the absolute value of a function

Quadrature of the absolute value of a function

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.