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SUMMARY A fast algorithm for solving systems of linear equations with banded Toeplitz matrices is studied. An important step in the algorithm is a novel method for the spectral factorization of the generating function associated with the Toeplitz matrix. The spectral factorization is extracted from the right deflating subspaces corresponding to the eigenvalues inside and outside the open unit disk of a companion matrix pencil constructed from the coefficients of the generating function. The factorization is followed by the Woodbury inversion formula and solution of several banded triangular systems. Stability of the algorithm is discussed and its performance is demonstrated by numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY The strong stability of a symplectic matrix is investigated from algorithmic and numerical viewpoints using a theory developed by S.K. Godunov. This theory is based on a different formulation of the Krein-Gelfand-Lidskii characterization of strong stability, better suited for numerical calculations. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we construct an approximate solution to large Sylvester equations of the form . The construction uses a new variant of the block Arnoldi algorithm which exploits the near-breakdowns, that is, the near singularities in the generated basis. As a consequence, the algorithm eliminates the directions which do not contribute to the approximate solution by keeping in the generated basis only the “active” vectors detected by a criterion based on the residual associated with the approximate solution. The effectiveness of the proposed algorithm is demonstrated on several examples, including the case where the matrix B has a small or a large size. [Copyright &y& Elsevier]