Computation of pseudospectra of matrices in parallel

## Project description

Mpseudo performs multicore and precise computation of pseudospectra of (square or rectangular) matricies. It uses pseudospectra definition and find epsilon-values on a regular grid of a complex plane. It uses multiprocessing module to share computations between cpu-cores, and mpmath module to make calculations with high precision.

## Dependencies

Mpmath module is needed to perform computations with high precision.

pip install mpmath

If you don’t need ability of high precision pseudospectra computation (more than 15 digits), the mpseudo can work without mpmath. The only requirement - NumPy. It should be installed on your system or in virtual environment.

## Installation

git clone https://github.com/scidam/mpseudo.git

## Example

The pseudospectrum of the gallery(5) MatLab matrix looks like this (up to 100-digits of accuracy used for a matrix resolvent computation):

Pseudospectrum of gallery(5) MatLab matrix

The pseudospectra above is obtained via the following lines of code:

from matplotlib import pyplot
from mpseudo import pseudo

# Gallery(5) MatLab matrix (exact eigenvalue is 0 (the only!))
A = [[-9, 11, -21, 63, -252],
[70, -69, 141, -421, 1684],
[-575, 575, -1149, 3451, -13801],
[3891, -3891, 7782, -23345, 93365],
[1024, -1024, 2048, -6144, 24572]]

# compute pseudospectrum in the bounding box [-0.05,0.05,-0.05,0.05] with
# resolution 100x100 (ncpu = 2 processes) and 50-digits precision.
psa, X, Y = pseudo(A, ncpu=2, digits=50, ppd=100, bbox=[-0.05,0.05,-0.05,0.05])

# show results
pyplot.conourf(X, Y, psa)
pyplot.show()


Note, if mpmath module is not installed, pseudospectrum of the matrix will be computed with standard (double, 15-digits) precision, which is not sufficient for this case.

Interesting, but Eigtool or PseudoPy tools (along with scipy eigvals function) applied to the matrix A in the example above lead to inaccurate results (due to insufficient (double) precision):

Pseudospectrum of gallery(5) MatLab matrix plotted via PseudoPy

## Project details

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