M-M.E.S.S. - The Matrix Equation Sparse Solver Library for MATLAB and Octave

M-M.E.S.S. provides low-rank solvers for large-scale symmetric matrix equations with sparse or sparse + low-rank coefficients. The main focus is on differential and algebraic Riccati equations appearing in control and model order reduction, as well as algebraic Lyapunov equations for, e.g., balanced truncation.

The underlying dynamical system may be of first or second order, and structured proper differential algebraic equations (DAEs) that allow for implicit index reduction are also supported.

The solvers philosophy is to always work on the implicitly linearized (for second order systems) and/or implicitly projected (in the DAE case) matrix equations. That means the implicit Lyapunov or Riccati equation is always of the form known for a standard first order ODE, that may have a non identity but invertible E matrix.

See help mess for an overview of supported matrix equations and system structures.

Further, M-M.E.S.S. provides functions for Balanced Truncation and (tangential) iterative rational Krylov algorithm (IRKA) for model order reduction (MOR) of first order state space systems and some examples demonstrate the use of the algorithms in MOR of second order systems and DAEs.

In close relation to the predecessor LyaPack, we use user supplied functions (usfs) that implement the actions of the system matrices E and A in multiplication and (shifted) solves. We provide those functions for standard state space systems, second order systems, structured DAEs of index 1 and 2, as well as second order DAEs of index 1, 2 and 3. For more information on usfs see help mess_usfs.

Copyright 2009-2023 by Jens Saak, Martin Koehler, Peter Benner (MPI Magdeburg)

The software uses a BSD 2-Clause license. See LICENSE.md and COPYING for details.

Installation Instructions

See INSTALL.md for details.

Getting started

Change to the installation directory, run mess_path and check help mess for the basic information about supported matrix equations and core solvers.

In case you need functionality beyond that of mess_lyap and mess_care, consult the demonstration routines in the DEMOS folder for example use cases of the other and underlying solvers.

Contact

WWW, GITLAB, GITHUB

email

Citation

See CITATION.md for details about citing the software.

Further reading

• P. Benner, M. Koehler, J. Saak, Matrix equations, sparse solvers: M-M.E.S.S.-2.0.1 – philosophy, features and application for (parametric) model order reduction, in: P. Benner, T. Breiten, H. Faßbender, M. Hinze, T. Stykel, R. Zimmermann (Eds.), Model Reduction of Complex Dynamical Systems, Vol. 171 of International Series of Numerical Mathematics, Birkhäuser, Cham, 2021, pp. 369–392. https://doi.org/10.1007/978-3-030-72983-7_18.
• J. Saak, M. Voigt, Model reduction of constrained mechanical systems in M-M.E.S.S., IFAC-PapersOnLine 9th Vienna International Conference on Mathematical Modelling MATHMOD 2018, Vienna, Austria, 21–23 February 2018 51 (2) (2018) 661–666. https://doi.org/10.1016/j.ifacol.2018.03.112.
• P. Benner, J. Saak, Efficient solution of large scale Lyapunov and Riccati equations arising in model order reduction problems, Proc. Appl. Math. Mech. 8 (1) (2008) 10085–10088. https://doi.org/10.1002/pamm.200810085.

History

• 2000 LyaPack: M-M.E.S.S. originates in the work of Penzl and especially his software package LyaPack.
• 2003-2007 LyaPack 1.1 - 1.8 authored by Jens Saak improve the handling of non-identity E matrices.
• 2008 the first conference talk about the new project labeled M.E.S.S. is held at GAMM 2008 in Bremen (Germany).
• 2016 M-M.E.S.S.-1.0 and 1.0.1 first public releases of the greatly rewritten toolbox.
• 2019 M-M.E.S.S.-2.0 adds differential Riccati equations.
• 2020 M-M.E.S.S.-2.0.1 fixes several bugs and adds improvements for MOR.
• 2021 M-M.E.S.S.-2.1 adds Lyapunov plus positive equations and BT of bilinear systems.
• 2022 M-M.E.S.S.-2.2 fixes several smaller bugs and adds improvements to code style and performance, and improves documentation
• M-M.E.S.S.-3.0 adds
  • Krylov-projection-based solvers
  • sparse-dense Sylvester equations

Roadmap

• M-M.E.S.S.-3.x

  • bilinear control problems
  • DAE usfs restructuring
  • consistency and efficiency improvements
  • code refactoring to avoid code duplication

• M-M.E.S.S.-4.0

  • sparse Sylvester equations
  • non-symmetric AREs