Matlab Programs in connection with the paper:

Hidden Positivity and a New Approach to

Numerical Computation of Hausdorff Dimension: Higher Order Methods

The following programs are used to compute the Hausdorff dimension of the Cantor set E_{a_1, a_2,...,a_R} consisting of all points x in the interval (0,1) with the continued fraction expansion: 

x = [a_1, a_2, a_3, ..., a_R] =                  1
                                        ----------------------
                                          a_1 +     1
                                                --------
                                                a_2 +    1
                                                     ---------
                                                     a_3 + ...
where each a_n is a positive integer.

hd1dmp.m This is the main program for the multiple precision version of approximation by piecewise polynomials of degree <= r using collocation at the extended Chebyshev points on each subinterval. It uses the Advanpix multiple precision toolbox.

eval1dPnitmp.m A program used by hd1dmp.m

thetabxitmp.m A program used by hd1dmp.m

thetabxitpmp.m A program used by hd1dmp.m

hd1dsp.m This is the main program for the single precision version of approximation by piecewise polynomials of degree <= r using collocation at the extended Chebyshev points on each subinterval.

eval1dPnitsp.m A program used by hd1dsp.m

thetabxitsp.m A program used by hd1dsp.m

thetabxitpsp.m A program used by hd1dsp.m

Matlab Programs in connection with the paper:

C^m Eigenfunctions of Perron-Frobenius Operators and a New Approach

to Numerical Computation of Hausdorff Dimension: Applications in R^1

The following programs are used to compute the Hausdorff dimension of the Cantor set E_{a_1, a_2,...,a_R} consisting of all points x in the interval (0,1) with the continued fraction expansion: 

x = [a_1, a_2, a_3, ..., a_R] =                  1
                                        ----------------------
                                          a_1 +     1
                                                --------
                                                a_2 +    1
                                                     ---------
                                                     a_3 + ...
where each a_n is a positive integer.

hd1dP1i.m This is the main program for approximation by continuous piecewise linear functions.

mat1d.m A program used by hd1dP1i.m

hd1dPni.m This is the main program for approximation by continuous piecewise polynomials of degree <= n.

eval1dPn.m A program used by hd1dPni.m

thetabx.m A program used by hd1dP1i.m and hd1dPni.m

The following programs are used to compute the Hausdorff dimension of the invariant set obtained from iterations of the mappings:
theta_1(x) = (1/(3+2*lam))*x*(1 + lam*x*x*sqrt(x))
theta_2(x) = (1/(3+2*lam))*x*(1 + lam*x*x*sqrt(x)) + (2+lam)/(3+2*lam)
where 0 <= lam <= 1.
The choice lam =0 corresponds to the middle thirds Cantor set, whose Hausdorff dimension is ln 2/ln 3.

hd1dperti.m The main program.

mat1dpert.m A program used by hd1dperti.m

thetabxlam.m A program used by hd1dperti.m


Matlab Programs in connection with the paper:

A New Approach to Numerical Computation of Hausdorff Dimension

of Iterated Function Systems: Applications to Complex Continued Fractions

The following programs are used to compute the Hausdorff dimension of several invariant sets in two dimensions.
For each of these problems, theta_b(z) = 1/(z+b) and z = x + iy belongs to
H:= {(x,y) in R^2: (x-1/2)^2 + y^2 <= 1/4, y >=0}.

The sets are:
I_1 = {b=m+ni: m is a positive integer and n an integer},
I_2 = {b=m-ni: m and n are positive integers},
I_3 = {b=m+ni: m = 1 and 2 and n = 0, 1, 2, -1,-2}.

hd2di.m The main program for the computation of the Hausdorff dimension of the three sets
given above using piecewise bilinear functions.

matas.m A program used by hd2di.m

thetabz.m A program used by hd2di.m, hd2dPni.m, and hd2dPnse.m.

kindx.m A program used by matas.m.

hd2dPni.m The main program for the computation of the Hausdorff dimension of the set I_3
given above using higher order piecewise polynomials.

eval2dPni.m A program used by hd2dPni.m.

hd2dPnse.m Computation of the Hausdorff dimension of a special example where the exact solution is known.

eval2dPnse.m A program used by hd2dPnse.m.

Maintained by falk@math.rutgers.edu and last modified 9/1/2017.