Matrix Equations PACKage (MEPACK)

Version: 1.1.1

Copyright 2017-2023 by Martin Köhler, MPI-Magdeburg

The software is licensed under GPLv3 or later. See LICENSE for details.

MEPACK is a Fortran software library for the solution of dense Sylvester-like matrix equations. This includes the following equations:

• Standard Sylvester Equation (SYLV) $AX\pm XB = Y$
• Standard Lyapunov Equation (LYAP) $AX + XA^T = Y$
• Standard Stein (Discrete-Time Lyapunov) Equation (STEIN) $AXA^T - X = Y$
• Discrete-Time Sylvester Equation (SYLV2) $AXB - X = Y$
• Generalized Sylvester Equation (GSYLV) $AXB \pm CXD = Y$
• Generalized Lyapunov Equation (GLYAP) $AXB^{T} + BXA^T = Y$
• Generalized Stein (Discrete-Time Lyapunov) Equation (GSTEIN) $AXA^T - EXE^T = Y$
• Generalized Coupled Sylvester Equation (CSYLV) $(AR \pm LB=E,CR \pm LD = F)$ and its dual counterpart (CSYLV_DUAL) $(AR + CL = E, \pm RB \pm LD =F)$.

The implemented solvers use a forward/backward substitution scheme on matrix equations with (quasi) triangular coefficient matrices. The algorithms are implemented in different ways, including the following features:

• Level-3 BLAS enabled solvers with adjustable block/tile sizes.
• OpenMP 4 accelerated Directed-Acyclic-Graph scheduled solvers with focus on current multicore CPUs
• Optimized BLAS level-2 solvers for problems with right-hand sides smaller than 128x128.
• Iterative refinement solvers to improve the accuracy if the (generalized) Schur decompositions are perturbed.

Besides that, the library implements some other solvers for performance comparison:

• The recursive blocking approach by Jonsson and Kågström,
• The Gardiner-Laub approach for the generalized Sylvester equation.

References for both techniques can be found below.

Installation

See Installation.

API Documentation

All routines in the src/ directory are documented in a BLAS/LAPACK style. The documentation can be extracted using doxygen during the build process. See Installation.

Configuration at Runtime

Some parameters, like the block sizes of the level-3, OpenMP 4, and the two stage algorithm, are hardware dependent and need to be tuned for an optimal runtime behavior. They can either be hard-coded or configured dynamically at runtime. The dynamic selection is used, if a block size or a parameter is set to zero at runtime. This is the default situation. The dynamic configuration is given as a LUA script, either at compile-time in the src/default_config.lua file or at runtime by passing a configuration file in the the MEPACK_LUA_CONFIG environment variable. Details about this feature can be found in Tuning.

In some cases, MEPACK prints some debug information to stderr. This can be controlled by setting the MEPACK_VERBOSE environment variable. A non-zero value means, that MEPACK shows some about the runtime environment, like the loaded configuration file. The computational routines are not effected by these outputs. Error while reading the configuration file will always be printed to the screen independent of the value of MEPACK_VERBOSE.

Using MEPACK your project

The MEPACK project exports CMake package files and pkg-config files to make MEPACK usable for other projects. The package files are located in the library directory in the installation prefix.

In order to use MEPACK with CMake it is included via

```cmake find_package(MEPACK REQUIRED)

or in case of 64 bit integers

find_package(MEPEACK64 REQUIRED)

... targetlinklibraries( yourtarget PRIVATE MEPACK::mepack ) ```

If the static version of MEPACK should be used, replace MEPACK::mepack with MEPACK::mepack_static.

To make the installed MEPACK project discoverable, add the MEPACK install prefix to the CMAKE_PREFIX_PATH variable. The usual install location of the package files is PREFIX/lib/cmake/mepack or PREFIX/lib/cmake/mepack64.

For non-CMake build systems (like make) you can use the exported pkg-config file by setting PKG_CONFIG_PATH to include the directory containing the exported pc-file. The usual install location of the pc-file is PREFIX/lib/pkgconfig. In make, you can obtain the required compile and link arguments with

make MEPACK_CFLAGS := $(shell pkg-config --cflags mepack) MEPACK_LIBS := $(shell pkg-config --libs mepack)

or

make MEPACK_CFLAGS := $(shell pkg-config --cflags mepack64) MEPACK_LIBS := $(shell pkg-config --libs mepack64)

if you require 64 bit integers.

Citation

If you are using MEPACK inside your project, please cite at least one of the following references: M. Köhler. 2021. Approximate Solution of Non-Symmetric Generalized Eigenvalue Problems and Linear Matrix Equation on HPC Platforms. Dissertation. Logos Verlag Berlin, Magdeburg, Germany. or M. Köhler. 2024, Matrix Equations PACKage -- A Fortran library for the solution of Sylvester-like Matrix equations, Zendo, DOI: 10.5281/zenodo.10568848

Examples

The examples directory provide a huge set of demonstration codes how to use MEPACK. Furthermore, these examples are also constructed to serve as tests for the library. Thus, most of them are executed when make test is run.

For details about the examples see examples/README.md.

External Codes included in the Examples

The examples contain a set of foreign codes for benchmark purpose. These codes are subject to the copyrights issued by their original authors. At the moment this effects the examples/slicot directory, which provides some routines from SLICOT (https://www.slicot.org, https://github.com/SLICOT/SLICOT-Reference, BSD-3 License): MB01RD, MB01RW, MB02UV, MB02UU, SB03MD, SB03MV, SB03MW, SB03MX, SB03MY, SB04PX, SB04PY, SG03AD, SG03AX, and SG03AY.

Furthermore, the codes of the following publications can be used for comparison in the examples: Judith D. Gardiner, Matthew R. Wette, Alan J. Laub, James J. Amato, and Cleve B. Moler. 1992. Algorithm 705; a FORTRAN-77 software package for solving the Sylvester matrix equation AXB_T_ + CXD_T_ = E. _ACM Trans. Math. Softw._ 18, 2 (June 1992), 232–238. DOI:https://doi.org/10.1145/146847.146930 and ``` Isak Jonsson and Bo Kågström. 2002. Recursive blocked algorithms for solving triangular systems—Part I: one-sided and coupled Sylvester-type matrix equations. ACM Trans. Math. Softw. 28, 4 (December 2002), 392–415. DOI:https://doi.org/10.1145/592843.592845

Isak Jonsson and Bo Kågström. 2002. Recursive blocked algorithms for solving triangular systems—Part II: two-sided and generalized Sylvester and Lyapunov matrix equations. ACM Trans. Math. Softw. 28, 4 (December 2002), 416–435. DOI:https://doi.org/10.1145/592843.592846 ```

Due to unclear licensing, these codes need to be provided by the user. See Installation for details.

Known Issues

• If the OpenBLAS 0.3.8 with PTHREADs is used, OMP_NUM_THREADS=1 needs to be set. The pthread scheduler of OpenBLAS conflicts with the one from OpenMP.
• The MATLAB interface ( not the Octave Interface ) cannot use OpenMP 4 accelerated solvers due to deadlocks in MATLAB's OpenMP translation interface.
• If doxygen 1.8.17 is installed on the system, like on Ubuntu 20.04, the documentation includes MathJax to render the formulas. This is due to a bug in 1.8.17 with some versions of Ghostscript, especially Ghostscript 9.50.-