Numerical Methods for Eigenvalue Problems

✍ Scribed by Steffen Börm; Christian Mehl

• Publisher: De Gruyter
• Year: 2012
• Tongue: English
• Leaves: 216
• Category: Library

✦ Synopsis

Eigenvalues and eigenvectors of matrices and linear operators play an important role when solving problems from structural mechanics and electrodynamics, e.g., by describing the resonance frequencies of systems, when investigating the long-term behavior of stochastic processes, e.g., by describing invariant probability measures, and as a tool for solving more general mathematical problems, e.g., by diagonalizing ordinary differential equations or systems from control theory.

This textbook presents a number of the most important numerical methods for finding eigenvalues and eigenvectors of matrices. The authors discuss the central ideas underlying the different algorithms and introduce the theoretical concepts required to analyze their behavior with the goal to present an easily accessible introduction to the field, including rigorous proofs of all important results, but not a complete overview of the vast body of research. Several programming examples allow the reader to experience the behavior of the different algorithms first-hand.

The book addresses students and lecturers of mathematics, physics and engineering who are interested in the fundamental ideas of modern numerical methods and want to learn how to apply and extend these ideas to solve new problems.

• Appropriate as a textbook for graduate courses or for independent study
• Several exercises and summaries of each chapter help beginners to understand the material
• Source codes of the presented algorithms available online

✦ Table of Contents

Preface
1 Introduction
1.1 Example: Structural mechanics
1.2 Example: Stochastic processes
1.3 Example: Systems of linear differential equations
2 Existence and properties of eigenvalues and eigenvectors
2.1 Eigenvalues and eigenvectors
2.2 Characteristic polynomials
2.3 Similarity transformations
2.4 Some properties of Hilbert spaces
2.5 Invariant subspaces
2.6 Schur decomposition
2.7 Non-unitary transformations
3 Jacobi iteration
3.1 Iterated similarity transformations
3.2 Two-dimensional Schur decomposition
3.3 One step of the iteration
3.4 Error estimates
3.5 Quadratic convergence
4 Power methods
4.1 Power iteration
4.2 Rayleigh quotient
4.3 Residual-based error control
4.4 Inverse iteration
4.5 Rayleigh iteration
4.6 Convergence to invariant subspace
4.7 Simultaneous iteration
4.8 Convergence for general matrices
5 QR iteration
5.1 Basic QR step
5.2 Hessenberg form
5.3 Shifting
5.4 Deflation
5.5 Implicit iteration
5.6 Multiple-shift strategies
6 Bisection methods
6.1 Sturm chains
6.2 Gershgorin discs
7 Krylov subspace methods for large sparse eigenvalue problems
7.1 Sparse matrices and projection methods
7.2 Krylov subspaces
7.3 Gram-Schmidt process
7.4 Arnoldi iteration
7.5 Symmetric Lanczos algorithm
7.6 Chebyshev polynomials
7.7 Convergence of Krylov subspace methods
8 Generalized and polynomial eigenvalue problems
8.1 Polynomial eigenvalue problems and linearization
8.2 Matrix pencils
8.3 Deflating subspaces and the generalized Schur decomposition
8.4 Hessenberg-triangular form
8.5 Deflation
8.6 The QZ step
Bibliography
Index