Numerical Methods for General and Structured Eigenvalue Problems
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Beschreibung
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].
Autor*in
Daniel Kressner
Themen in »Numerical Methods for General and Structured Eigenvalue Problems«
algorithms computational methods eigenvalue matrix product structured matrix
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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)
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