readme.md

This is a repository containing tools to rearrange two-variables functions in MATLAB.

Installation

Add the folder "Code" to your MATLAB path and you are ready to go!

Usage

Fucntions are stored as contourLines objects. This is a class that contains all the needed informations to rearrange.

Constructor

Given a two-variables function F, you want to pass it to the constructor as a function handle.

function z = F(x,y)
	z=x+y;
end
Fsharp = contourLines(@F);

The line Fsharp = contourLines(@F); will create a contourLines object named Fsharp, which stores 100 levels of the function F . The complete constructor method can be called as follows.

Fsharp = contourLines(@F, gridCoordinates,numberOfCells,numberOfLevels);

When specified, the variables should be of the following form:

• gridCoordinates=[xmin xmax ymin ymax] is describing the square domain in which the function is restricted to; default [-1 1 -1 1];
• numberOfCells determines how many points are constructed in the grid; default 20 will give a 20x20 grid;
• numberOfLevels determines how many levels of the function are stored; default 100

Different bounded domains

If you want to construct a function with a domain E different from a rectangle, you can pass to the constructor method a function F which is -inf outside of E.

Schwarz rearrangement

Given a function F and the contourLines object Fsharp relative to it, you can call the Schwarz (spherically decreasing) rearrangement of F as the function

Fsharp.Schwarz(x,y);

with $x$ and $y$ real numbers. If the couple $(x,y)$ is outside of the ball where the Schwarz rearrangement is defined, a NaN will be given as an output.

Rearrangement along another function

An example is given in the file Rearrangement2.m.

Given two functions F, G and the respective contourLines objects Fsharp and Gsharp, you can call the rearrangement of F along G as the function

Fsharp.Schwarz(x,y,Gsharp);

with x and y real numbers. If the couple (x,y) is outside of the set where the Schwarz rearrangement is defined (i.e. the biggest superlevel set of G inside the initial grid), a NaN will be given as an output.

Plot Level

Given a contourLines object Fsharp, you can plot the i-th level of Fsharp with the line

Fsharp.plotLevel(i)

The number i can go from 1 to Fsharp.numberOfLevels. Bigger the i, smaller the level set.

Distribution function

The distribution function of $F$ is the function

$$\mu_F(t)=\lvert \set{f>t}\rvert.$$

A distribution function is stored inside the class contourLines. If Fsharp is a contourLines object, you can call its distribution function by

Fsharp.distributionFunction(t);

where t is a real number.

Radially decreasing rearrangement

The radially decreasing rearrangement of $F$ is defined as

$$F^*(m)=\inf \Set{t\in\mathbb{R} | \mu_F(t)\le m}.$$

The radially decreasing rearrangement function is stored in the contourLines object, and you can call it by

Fsharp.radDecrRearr(m);

where m is a real number.