Algorithms for Quadratic Matrix and Vector Equations
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 equa
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tesi di perfezionamento in Matematica sostenuta il 15 dicembre 2010 C OMMISSIONE G IUDICATRICE Stefano Marmi, Presidente Dario Bini Giuseppe Da Prato Roberto Frigerio Patrizia Gianni Beatrice Meini Andrea Carlo Giuseppe Mennucci
Federico Poloni Sekretariat MA 4-5 Straße des 17. Juni 136 D-10623 Berlin Algorithms for Quadratic Matrix and Vector Equations
Federico Poloni
Algorithms for Quadratic Matrix and Vector Equations
c 2011 Scuola Normale Superiore Pisa
ISBN 978-88-7642-383-3 ISBN 978-88-7642-384-0 (eBook)
Contents
Introduction
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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. Inde1nite product spaces . . . . . . . . . . . . 1.7. M¨obius transformations and Cayley transforms 1.8. Control theory terminology . . . . . . . . . . . 1.9. Eigenvalue splittings . . . . . . . . . . . . . .
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Part I – Quadratic vector and matrix equations 2 Quadratic vector equations 2.1. Introduction . . . . . . . . . . . . . . . . 2.2. General problem . . . . . . . . . . . . . 2.3. Concrete cases . . . . . . . . . . . . . . 2.4. Minimal solution . . . . . . . . . . . . . 2.4.1. Existence of the minimal solution 2.4.2. Taylor expansion . . . . . . . . . 2.4.3. Concrete cases . . . . . . . . . . 2.5. Functional iterations . . . . . . . . . . . 2.5.1. De1nition and convergence . . . . 2.5.2. Concrete cases . . . . . . . . . . 2.6. Newton’s method . . . . . . . . . . . . . 2.6.1. De1nition and convergence . . . . 2.6.2. Concrete cases . . . . . . . . . . 2.7. Modi1ed Newton method . . . . . . . . .
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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 . . . . .
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3 A Perron vector iteration for QVEs 3.1. Applications . . . . . . . . . . . . . . . . . 3.2. Assumptions on the problem . . . . . . . . 3.3. 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.
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