Function Repository Resource:

MandelbrotMatrix

Construct a matrix whose eigenvalues lie on the Mandelbrot set boundary

Contributed by: Jan Mangaldan

ResourceFunction["MandelbrotMatrix"][n]

gives the n^th Mandelbrot matrix.

Details

The Mandelbrot matrix is a sparse upper Hessenberg matrix whose eigenvalues lie on the boundary of the Mandlebrot set.

ResourceFunction["MandelbrotMatrix"] returns a SparseArray.

ResourceFunction["MandelbrotMatrix"][n] gives a matrix of dimensions (2ⁿ⁺¹-1)×(2ⁿ⁺¹-1).

A Mandelbrot matrix:

In[1]:=

Out[1]=

Plot its eigenvalues in the complex plane:

In[2]:=

Out[2]=

Visualize the eigenvalues of a large Mandelbrot matrix:

In[3]:=

Out[3]=

Citation:
- Chan E.Y.S., "A comparison of solution methods for Mandelbrot-like polynomials." Western University Electronic Thesis and Dissertation Repository, 2016.