Umfang:
Online-Ressource (250p, digital)
ISBN:
9788876423840
Serie:
Tesi/Theses 16
Inhalt:
This book is devoted to studying algorithms for the solution of a class of quadratic matrix and vector equations. These equations appear, in different forms, in several practical applications, especially in applied probability and control theory. The equations are first presented using a novel unifying approach; then, specific numerical methods are presented for the cases most relevant for applications, and new algorithms and theoretical results developed by the author are presented. The book focuses on “matrix multiplication-rich” iterations such as cyclic reduction and the structured doubling algorithm (SDA) and contains a variety of new research results which, as of today, are only available in articles or preprints.
Anmerkung:
Description based upon print version of record
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Auf Homepage von Springer unter der Reihe "Publications of the Scuola Normale Superiore" aufgeführt
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Title Page; Copyright Page; Table of Contents; Introduction; Chapter 1 Linear algebra preliminaries; 1.1. Nonnegative matrices and M-matrices; 1.2. Sherman-Morrison-Woodbury formula; 1.3. Newton's method; 1.4. Matrix polynomials; 1.5. Matrix pencils; 1.6. Indefinite product spaces; 1.7. Möbius transformations and Cayley transforms; 1.8. Control theory terminology; 1.9. Eigenvalue splittings; Part I Quadratic vector and matrix equations; Chapter 2 Quadratic vector equations; 2.1. Introduction; 2.2. General problem; 2.3. Concrete cases; 2.4. Minimal solution
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2.4.1. Existence of the minimal solution2.4.2. Taylor expansion; 2.4.3. Concrete cases; 2.5. Functional iterations; 2.5.1. Definition and convergence; 2.5.2. Concrete cases; 2.6. Newton's method; 2.6.1. Definition and convergence; 2.6.2. Concrete cases; 2.7. Modified Newton method; 2.7.1. Theoretical properties; 2.7.2. Concrete cases; 2.8. Positivity of the minimal solution; 2.8.1. Role of the positivity; 2.8.2. Computing the positivity pattern; 2.9. Other concrete cases; 2.10. Conclusions and research lines; Chapter 3 A Perron vector iteration for QVEs; 3.1. Applications
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3.2. Assumptions on the problem3.3. The optimistic equation; 3.4. The Perron iteration; 3.5. Convergence analysis of the Perron iteration; 3.5.1. Derivatives of eigenvectors; 3.5.2. Jacobian of the Perron iteration; 3.5.3. Local convergence of the iteration; 3.6. Numerical experiments; 3.7. Conclusions and research lines; Chapter 4 Unilateral quadratic matrix equations; 4.1. Applications; 4.2. Overview of logarithmic and cyclic reduction; 4.3. Generalization attempts; Chapter 5 Nonsymmetric algebraic Riccati equations; 5.1. Applications; 5.1.1. Application to fluid queues
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5.1.2. Application to transport equation5.2. Theoretical properties; 5.2.1. The dual equation; 5.2.2. Existence of nonnegative solutions; 5.2.3. The eigenvalue problem associated with the matrix equation; 5.2.4. The Fréchet derivative of the Riccati operator; 5.2.5. The number of positive solutions; 5.2.6. Perturbation analysis for the minimal solution; 5.3. Schur method; 5.4. Functional iterations and Newton's method; 5.5. Matrix sign method; 5.6. Block swapping in pencil arithmetic; 5.7. Inverse-free NMS iteration; 5.8. Matrix sign and disk iteration; 5.9. The structured doubling algorithm
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5.9.1. Standard structured form5.9.2. Derivation of SDA; 5.9.3. Convergence results; 5.9.4. A new method to compute the SSF-I of a pencil; 5.9.5. SDA for nonsymmetric algebraic Riccati equations; 5.10. Conclusions and research lines; Chapter 6 Transforming NAREs into UQMEs; 6.1. Introduction; 6.2. Assumptions on Algebraic Riccati Equations; 6.3. Transformations of a NARE into a UQME; 6.3.1. Ramaswami's transformation; 6.3.2. UL-based transformation; 6.4. Eigenvalue transformations; 6.4.1. Shrink and shift; 6.4.2. Cayley transform; 6.5. Old and new algorithms
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6.5.1. Shrink-and-shift with Ramaswami's transformation
Weitere Ausg.:
ISBN 9788876423833
Weitere Ausg.:
Buchausg. u.d.T. Poloni, Federico, 1983 - Algorithms for quadratic matrix and vector equations Pisa : Ed. della Normale, 2011 ISBN 9788876423833
Weitere Ausg.:
ISBN 8876423834
Sprache:
Englisch
Fachgebiete:
Mathematik
DOI:
10.1007/978-88-7642-384-0
URL:
Volltext
(lizenzpflichtig)
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