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Numerical Methods for General and Structured Eigenvalue Problems

By: Kressner, Daniel [author.].
Contributor(s): SpringerLink (Online service).
Material type: materialTypeLabelBookSeries: Lecture Notes in Computational Science and Engineering: 46Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2005.Description: XIV, 258 p. 32 illus. online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9783540285021.Subject(s): Mathematics | System theory | Computer mathematics | Mathematics | Computational Mathematics and Numerical Analysis | Systems Theory, Control | Computational Science and EngineeringDDC classification: 518 Online resources: Click here to access online
Contents:
The QR Algorithm -- The QZ Algorithm -- The Krylov-Schur Algorithm -- Structured Eigenvalue Problems -- Background in Control Theory Structured Eigenvalue Problems -- Software.
In: Springer eBooksSummary: 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].
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The QR Algorithm -- The QZ Algorithm -- The Krylov-Schur Algorithm -- Structured Eigenvalue Problems -- Background in Control Theory Structured Eigenvalue Problems -- Software.

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].

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