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. Read more... 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
Author(s): Steffen Borm; Christian Mehl
Series: De Gruyter textbook
Publisher: De Gruyter
Year: 2012
Language: English
Pages: 217
City: Berlin
Tags: Математика;Вычислительная математика;
Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 8
1.1 Example: Structural mechanics......Page 10
1.2 Example: Stochastic processes......Page 13
1.3 Example: Systems of linear differential equations......Page 14
2.1 Eigenvalues and eigenvectors......Page 17
2.2 Characteristic polynomials......Page 21
2.3 Similarity transformations......Page 24
2.4 Some properties of Hilbert spaces......Page 28
2.5 Invariant subspaces......Page 33
2.6 Schur decomposition......Page 35
2.7 Non-unitary transformations......Page 42
3.1 Iterated similarity transformations......Page 48
3.2 Two-dimensional Schur decomposition......Page 49
3.3 One step of the iteration......Page 52
3.4 Error estimates......Page 56
3.5 Quadratic convergence......Page 62
4.1 Power iteration......Page 70
4.2 Rayleigh quotient......Page 75
4.3 Residual-based error control......Page 79
4.4 Inverse iteration......Page 82
4.5 Rayleigh iteration......Page 86
4.6 Convergence to invariant subspace......Page 88
4.7 Simultaneous iteration......Page 92
4.8 Convergence for general matrices......Page 100
5.1 Basic QR step......Page 109
5.2 Hessenberg form......Page 113
5.3 Shifting......Page 122
5.4 Deflation......Page 125
5.5 Implicit iteration......Page 127
5.6 Multiple-shift strategies......Page 135
6 Bisection methods......Page 141
6.1 Sturm chains......Page 143
6.2 Gershgorin discs......Page 150
7.1 Sparse matrices and projection methods......Page 154
7.2 Krylov subspaces......Page 158
7.3 Gram-Schmidt process......Page 161
7.4 Arnoldi iteration......Page 168
7.5 Symmetric Lanczos algorithm......Page 173
7.6 Chebyshev polynomials......Page 174
7.7 Convergence of Krylov subspace methods......Page 181
8.1 Polynomial eigenvalue problems and linearization......Page 191
8.2 Matrix pencils......Page 194
8.3 Deflating subspaces and the generalized Schur decomposition......Page 198
8.4 Hessenberg-triangular form......Page 201
8.5 Deflation......Page 205
8.6 The QZ step......Page 207
Bibliography......Page 212
Index......Page 215
