Journal ArticleDOI
Numerical Determination of a Canonical Form of a Symplectic Matrix
S. K. Godunov,M. Sadkane +1 more
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TLDR
In this article, the authors proposed an algorithm that transforms a real symplectic matrix with a stable structure to a block diagonal form composed of three main blocks, which satisfy a modification of the Krein-Gelfand-Lidskii criterion.Abstract:
We propose an algorithm that transforms a real symplectic matrix with a stable structure to a block diagonal form composed of three main blocks. The two extreme blocks of the same size are associated respectively with the eigenvalues outside and inside the unit circle. Moreover, these eigenvalues are symmetric with respect to the unit circle. The central block is in turn composed of several diagonal blocks whose eigenvalues are on the unit circle and satisfy a modification of the Krein-Gelfand-Lidskii criterion. The proposed algorithm also gives a qualitative criterion for structural stability.read more
Citations
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Journal ArticleDOI
Optimal symplectic Householder transformations for SR decomposition
TL;DR: This work introduces symplectic Householder transformations and shows their main features and constructs a new algorithm, the analogous of the classical QR factorization, via Householders transformations, which involves free parameters.
Journal ArticleDOI
Spectral Analysis of Symplectic Matrices with Application to the Theory of Parametric Resonance
S. K. Godunov,M. Sadkane +1 more
TL;DR: The numerical analysis of the spectral structure of symplectic matrices and linear Hamiltonian systems with periodic coefficients and applications to the theory of parametric resonance are illustrated by spectral portraits calculation.
Journal ArticleDOI
Equivalence between modified symplectic Gram-Schmidt and Householder SR algorithms
A. Salam,E. Al-Aidarous +1 more
TL;DR: The SR factorization for a given matrix A is a QR-like factorization A=SR, where the matrix S is symplectic and R is J-upper triangular, and the MSGS is the symplectic Gram-Schmidt algorithm implemented via the SGS.
Journal ArticleDOI
On the strong stability of symplectic matrices
Mouhamadou Dosso,Miloud Sadkane +1 more
TL;DR: The strong stability of a symplectic matrix is investigated from algorithmic and numerical viewpoints using a theory developed by S.K. Godunov based on a different formulation of the Krein-Gelfand-Lidskii characterization of strong stability better suited for numerical calculations.
Journal ArticleDOI
A spectral trichotomy method for symplectic matrices
Mouhamadou Dosso,Miloud Sadkane +1 more
TL;DR: This paper presents an algorithm for the numerical approximation of spectral projectors onto the invariant subspaces corresponding to the eigenvalues inside, on, and outside the unit circle of a symplectic matrix.
References
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The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
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TL;DR: In this paper, the authors describe the historical development of the classical theory of linear methods for solving nonstiff ODEs and present a modern treatment of Runge-Kutta and extrapolation methods.
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Algebraic Riccati equations
Peter Lancaster,Leiba Rodman +1 more
TL;DR: Geometric theory: the complex case 8.
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Spectral Analysis of Symplectic Matrices with Application to the Theory of Parametric Resonance
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