Numerical Methods for General and Structured Eigenvalue Problems

1. Front Matter

   Pages I-XIV

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2. The QR Algorithm

   Pages 1-66

3. The QZ Algorithm

   Pages 67-111

4. The Krylov-Schur Algorithm

   Pages 113-130

5. Structured Eigenvalue Problems

   Pages 131-214

6. Background in Control Theory Structured Eigenvalue Problems

   Pages 215-223

7. Software

   Pages 225-231

8. Back Matter

   Pages 233-264

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The purpose of this book is to describe recent developments in solving eig- value problems, in particular with respect to the QR and QZ algorithms as well as structured matrices. Outline Mathematically speaking, the eigenvalues of a square matrix A are the roots of its characteristic polynomial det(A??I). An invariant subspace is a linear subspace that stays invariant under the action of A. In realistic applications, it usually takes a long process of simpli?cations, linearizations and discreti- tions before one comes up with the problem of computing the eigenvalues of a matrix. In some cases, the eigenvalues have an intrinsic meaning, e.g., for the expected long-time behavior of a dynamical system; in others they are just meaningless intermediate values of a computational method. The same applies to invariant subspaces, which for example can describe sets of initial states for which a dynamical system produces exponentially decaying states. Computing eigenvalues has a long history, dating back to at least 1846 when Jacobi [172] wrote his famous paper on solving symmetric eigenvalue problems. Detailed historical accounts of this subject can be found in two papers by Golub and van der Vorst [140, 327].

• algorithms
• computational methods
• eigenvalue
• matrix product
• structured matrix

From the reviews:

"This book deals with the numerical methods for general and structured eigenvalue problems. … These lecture notes provide a detailed treatment and an updated account on the various eigenvalue algorithms and related topics … . In today’s world of computing, it is certainly a valuable reference and tool for anyone using methods of numerical linear algebra." (Amin Boumenir, Zentralblatt MATH, Vol. 1079, 2006)

“A detailed treatment of methods for special matrices such as unitary, skew-Hamiltonian, Hamiltonian, symplectic, and product form. … Kressner has two very useful short appendices, one on basic ideas in control theory and the other on available software. … present the state of the art, for general and special eigenvalue problems, in a very clear manner. … That said, we are lucky to have … such fine books that reveal the clever ideas and techniques that constitute current matrix eigenvalue algorithms.” (Beresford Parlett, SIAM Review, Vol. 52 (4), 2010)

• Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

  Daniel Kressner