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Computation of pseudospectra of matrices in parallel

Project description

# Mpseudo [![Build Status](https://travis-ci.org/scidam/mpseudo.svg?branch=master)](https://travis-ci.org/scidam/mpseudo)

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](http://numpy.org). 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](gal5pseudo.png)

The pseudospectra above is obtained via the following lines of code:
```python
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](http://www.cs.ox.ac.uk/pseudospectra/eigtool/) or [PseudoPy](https://github.com/andrenarchy/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](inacpseudo.png)

Read about this script in Russian [here](http://easydan.com/arts/arch/high-prec-pseudospectra/).

## License
Mpseudo is free software licensed under the [MIT License](http://opensource.org/licenses/mit-license.php).

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