RealPSPA implements in MATLAB, the criss-cross algorithm for computing the real \(\varepsilon\)–pseudospectral abscissa of a real matrix \(A\in {\mathbb R}^{n\times n}\):
\[ \alpha_\varepsilon^{\mathbb R}(A) =\max \left\{\mbox{Re}(\lambda)\, \colon\, \lambda \in \Lambda_{\varepsilon}^\mathbb R(A)\right\}, \]
where
\[ \Lambda_{\varepsilon}^ \mathbb R (A) = \left\{\lambda \in \mathbb C: \lambda \in \Lambda(A+E) \mbox{ with }E\in \mathbb R^{n\times n}, \|E\|_2\leq \varepsilon \right\}, \]
and \(\Lambda(A)\) denotes the eigenvalues of a matrix.
This package contains functions suitable for both dense and large sparse matrices \(A\). A detailed description of the algorithms can be found in the paper Ref. [1].
Ding Lu (University of Geneva)
Bart Vandereycken (University of Geneva)
Main functions:
Package download link RealPSPA_v01.tar.gz — (updated on June 13th, 2016).
Remarks: The code should work for any (relatively recent) version of MATLAB. It was tested on 2013b, 2016a (LINUX). We encourage users to run the code with 2016a, due to its new features in the svds function called by rpsas.
Data and demo routines for numerical examples in Ref. [1].
Download link rpsa_examples.tar.gz — (updated on June 13th, 2016).
Remarks: For a few examples, we need EigTool for data matrices generating, and the low–rank dynamical algorithm (Rostami's) for performance comparison. Please download these packages using the links below.
Rostami's Algorithm: http://users.wpi.edu/~mwu/webpage_code/pseudo/
Permission to use, copy,and modify this software for any purpose and without fee is hereby granted. For publications that uses this software, a reference to the paper, Ref. [1] below, will be appreciated.
See Ref. [1] for details.
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Fig. I.a: Demmel 3–by–3 matrix, \(\varepsilon=10^{-3.2}\).
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Fig. I.b: Demmel 5–by–5 matrix, \(\varepsilon=10^{-2}\). First 3 iterations, same figure setting as above. |
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Fig. II.a: Grcar(100) matrix, \(\varepsilon=0.3\), iterations k= 1, 2, 4.
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Fig. II.b: \(-\)Grcar(100) matrix, \(\varepsilon=0.2\), iterations k= 1, 2, 5. Same figure setting as above. |
Criss-cross type algorithms for computing the real pseudospectral abscissa
by Ding Lu
and Bart Vandereycken
SIAM J. Matrix Anal. Appl., 2017. 38(3):891–923.
(paper)
Released June 13, 2016
Email: Ding.Lu@unige.ch
Homepage: http://www.unige.ch/~dlu