Antoulas, A. C.; Sorensen, D. C. Lyapunov, Lanczos, and inertia. (English) Zbl 0997.65065 Linear Algebra Appl. 326, No. 1-3, 137-150 (2001). The authors present a method for solving a Lyapunov equation, and hence computing the inertia of a matrix, directly using \(\mathcal{O}(n^{3})\) floating point operations, without computing eigenvalues. This method is based on a specialized Lanczos process which reduces a matrix to Schwarz form. Unfortunately this method is numerically unstable and is mainly of theoretical interest, as the authors point out. However it is interesting because it develops some stimulating ideas about the hard problem of computing the inertia of a matrix without computing eigenvalues. Reviewer: Raffela Pavani (Milano) Cited in 2 Documents MSC: 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 15A24 Matrix equations and identities 65F30 Other matrix algorithms (MSC2010) Keywords:Lanczos method; Lyapunov equation; inertia; stability Software:Algorithm 432 PDFBibTeX XMLCite \textit{A. C. Antoulas} and \textit{D. C. Sorensen}, Linear Algebra Appl. 326, No. 1--3, 137--150 (2001; Zbl 0997.65065) Full Text: DOI References: [1] Anderson, B. D.O.; Jury, E. I.; Mansour, M., Schwarz matrix properties for continuous and discrete time systems, Int. J. Control, 23, 1-16 (1976) · Zbl 0336.93022 [2] A.C. Antoulas, Lectures on the approximation of large-scale dynamical systems, Department of Electrical and Computer Engineering, Rice University, 2000. To appear, SIAM, Philadelphia, PA; A.C. Antoulas, Lectures on the approximation of large-scale dynamical systems, Department of Electrical and Computer Engineering, Rice University, 2000. 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