StratiGraph is a Java-based tool for computing and displaying the stratification (closure hierarchy) of orbits or bundles of canonical structures. Matrix Canonical Structure (MCS) Toolbox is a Matlab toolbox for computing and representing canonical structure information.
Alan Edelman (MIT, Massachusetts) and Paul Van Dooren (UCL, Belgium)
The determination of the canonical form (Jordan, Kronecker, etc.) of a matrix or matrix pencil is an ill-posed problem in the presence of roundoff errors when the matrix or matrix pencil has multiple defective or derogatory eigenvalues. Therefore there exists modern numerical software based on staircase algorithms, such as GUPTRI, that regularizes these problems by allowing a tolerance for rank decisions to find their structure.
However, the algorithms used are known to occasionally fail and thereby accidentally producing wrong, but nearby structures. Failure appears to occur when the matrix or matrix pencil is close to a manifold of interesting structures of higher codimension.
Alan Edelman, Erik Elmroth, and Bo Kågström have proposed to make use of the mathematical knowledge of stratification of the obits or bundles of the canonical structures in order to enhance the staircase algorithm. This stratification, in effect, shows which structures are nearby other structures (in the sense of being in the closure) in the space of matrices. The stratification can be described as a connected graph, that grows exponentially with increasing matrix dimension.
StratiGraph is a Java-based software tool that determines and visualizes closure hierarchy graphs associated with orbit and bundle stratifications.
Current version of StratiGraph supports stratification of: • matrices under similarity, • matrix pencils G-sH under strict equivalence, • controllability pairs (A,B) under feedback equivalence, • observability pairs (A,C) under injection equivalence, • non-singular generalized state-space systems under feedback-injection equivalence, and • linearizations of (full normal-rank) matrix polynomials.
Matrix Canonical Structure Toolbox
The Matrix Canonical Structure (MCS) Toolbox for Matlab provides a framework with data type objects for representing canonical structures and computational functions related to canonical forms. Together with a plug-in to StratiGraph it is possible to import and export canonical structures between StratiGraph and Matlab.
Current version of MCS Toolbox supports canonical structures of: • matrices (under similarity, congruence, and *congruence), • matrix pencils (under strict equivalence, and symmetric and skew-symmetric pencils under congruence), • matrix polynomials, • system pencils (under feedback-injection equivalence).
The MCS toolbox includes routines for computing the canonical structure information using staircase algorithms. These are based on the GUPTRI (Generalized UPper TRIangular) algorithm [Demmel & Kågström, 1993] for matrix pencils. The GUPTRI form reveals the fine canonical structure information of a matrix pencil. For example, for a linearized model we can compute its canonical structure and then let StratiGraph determine and visualize nearby structures in the closure hierarchy.
StratiGraph 4.0 (2018-04-19) Requires Java version 7 or later. For changes see the change log. To run StratiGraph together with the MCS Toolbox, at least Matlab version 8.2 (R2013b) and the additional Java API matlabcontrol.jar are needed. See the readme file how to install and run.
MCS Toolbox 0.7 (2019-09-13) For changes see the change log. For full functionallity with StratiGraph the interface files (download here) in the Matlab plugin for StratiGraph 4.0 must be updated. See included readme file.
A. Edelman, E. Elmroth, and B. Kågström. A geometric approach to perturbation theory of matrices and matrix pencils. Part II: A stratification-enhanced staircase algorithm. SIAM J. Matrix Anal. Appl., vol. 20(3), pp. 667-669, 1999.
A. Edelman, E. Elmroth, and B. Kågström. A geometric approach to perturbation theory of matrices and matrix pencils. Part I: Versal deformations. SIAM J. Matrix Anal. Appl., vol. 18(3), pp. 653-692, 1997.
B. Kågström, S. Johansson, and P. Johansson. StratiGraph Tool: Matrix Stratification in Control Applications. In L. Biegler, S. Campbell, and V. Mehrmann, editors, Control and Optimization with Differential-Algebraic Constraints, ch. 5. SIAM Publications, 2012. (preprint)