pymepack 1.1.1

A wrapper for the MEPACK (Matrix Equation Package) library

python-mepack (pyMEPACK)

Version: 1.1.1

Copyright 2023-2024 by Martin Köhler, MPI-Magdeburg

Description

Python Interface for MEPACK

• a Fortran software library for the solution of dense Sylvester-like matrix equations.

The pyMEPACK interfaces routines solving the following equations:

• Standard Lyapunov equations (gelyap) $AX + XA^T = Y$
• Standard Stein (Discrete-Time Lyapunov) equations (gestein)$AXA^T - X = Y$
• Generalized Lyapunov equations (gglyap) $AXB^T + BXA^T = Y$
• Generalized Stein (Discrete-Time Lyapunov) Equation (ggstein) $AXA^T - BXB^T = Y$
• Standard Sylvester equations (gesylv) $AX + XB = Y$
• Discrete-time Sylvester equations (gesylv2) $AXB + X = Y$
• Generalized Sylvester equations (ggsylv) $AXB + CXD = Y$
• Generalized coupled Sylvester equations (ggcsylv) $AR + LB = E, CR + LD = F$
• Dual generalized coupled Sylvester equations (ggcsylv_dual) $AR + CL = E, RB + LD = F$

The library includes single and double precision solvers with iterative refinement for the above equations.

Dependencies

To install and run pyMEPACK the following components are required:

• MEPACK Version 1.1.1
• a BLAS and LAPACK implementation
• Python 3.7.0 +
• Cython 0.29.28 +
• numpy 1.20.0 +
• scipy 1.6.0 +
• setuptools 59.0.0 +
• configparser 5.2.0 +
• parameterized 0.8.0 + (only for tests)
• h5py 3.6.0 + (only for benchmarks)
• slycot 0.4.0 (only for benchmarks)
• Sphinx 5.0.2 + (for documentation)

Installation

pyMEPACK requires MEPACK to be installed on your system. See MEPACK's installation guide for detail (https://gitlab.mpi-magdeburg.mpg.de/software/mepack-release/-/blob/master/doc/install.md?ref_type=heads).

If MEPACK is not installed in a default location or the BLAS and LAPACK library are not named blas and lapack the pymepack.cfg file can be used to setup these differences. See pymepack.cfg-sample for details.

The installation of pyMEPACK is done by executing the following commands in the root directory of the project:

pip install .

or

pip install --user .

After a successful installation, pyMEPACK can be imported via import pymepack.

Documentation

Documentation of the pyMEPACK functions is accessible in the form of __doc__ strings.

HTML Documentation can be build with Sphinx inside the docs directory:

(cd ./docs && make html)

How to use pyMEPACK

The interface of pyMEPACK is very concise and easy to work with. The following code snippet solves a Lyapunov equation and computes the relative residual of the solution.

#!/usr/bin/env python3

import pymepack as pme
import numpy as np
import scipy as sp

n = 1000

# Prepare
A = np.triu(np.ones((n,n))) + np.diag(np.ones((n)))
X = np.ones((n,n))
Y = A @ X + X @ A.conj().T

# Solve
Xcomp, *_ = pme.gelyap(A, Y)

# Compute the residual
RelRes = pme.res_gelyap(A, Xcomp, Y)

print("Size = {:d} RelRes = {:e}".format(n, RelRes))

Testing

pyMEPACK contains a test suite. This is executed via

(cd /tmp && python3 -m unittest -v pymepack.tests)

The test suite cannot run from the root of source code after installation.

Test data

gelyap , gglyap , gestein and ggstein solvers, as well as their respective versions with iterative refinement are tested on examples provided in SLICOT benchmark collections, namely BB03AD and BB04AD [1,2].

All the Sylvester solvers are tested using randomization. We use numpy.random module and supply the random generator with a seed to avoid build instability. Last tested with numpy.version = 1.24.1. The non-symmetric stable matrices are constructed as in the example 4.1 in [1].

Tuning and Benchmarks

See Benchmark and Tuning.

License

pyMEPACK, like MEPACK it self, is license under GPLv3.

Authors

• Martin Köhler, MPI Magdeburg
• Aleksey Maleyko, MPI Magdeburg

Citation

Martin Köhler. (2024). MEPACK: Matrix Equation PACKage (1.1.1). Zenodo. https://doi.org/10.5281/zenodo.10568848

References

[1] Benner, P., 2004. Factorized solution of Sylvester equations with applications in control. sign (H), 1, p.2.

[2] D. Kressner, V. Mehrmann, and T. Penzl. CTLEX - a Collection of Benchmark Examples for Continuous-Time Lyapunov Equations. SLICOT Working Note 1999-6, 1999.

[3] D. Kressner, V. Mehrmann, and T. Penzl. DTLEX - a Collection of Benchmark Examples for Discrete-Time Lyapunov Equations. SLICOT Working Note 1999-7, 1999.