Limit search to available items
Book Cover
Author Börm, Steffen

Title Numerical Methods for Eigenvalue Problems
Published Berlin : De Gruyter, 2012


Description 1 online resource (216 pages)
Series De Gruyter Textbook
De Gruyter textbook.
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 convergence4 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 subspaces7.3 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
Summary 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 behaviour. Several programming examples allow the reader to experience the behaviour of the different algorithms first-hand. The book addresses students and lecturers of mathematics 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 ne
Analysis Bisection Method
Eigenvalue Problem
Jacobi Iteration
QR Iteration
Vector Iteration
Notes Print version record
Subject Eigenvalues.
Matrices -- Data processing
MATHEMATICS -- Algebra -- Elementary.
Matrices -- Data processing
Numerisches Verfahren
Form Electronic book
Author Mehl, Christian, 1968-
ISBN 9783110250374