Numerical Methods for Eigenvalue Problems (eBook)

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Steffen Börm, Christian-Albrechts-Universität zu Kiel, Germany; Christian Mehl, Technische Universität Berlin, Germany.

Preface 5
1 Introduction 9
1.1 Example: Structural mechanics 9
1.2 Example: Stochastic processes 12
1.3 Example: Systems of linear differential equations 13
2 Existence and properties of eigenvalues and eigenvectors 16
2.1 Eigenvalues and eigenvectors 16
2.2 Characteristic polynomials 20
2.3 Similarity transformations 23
2.4 Some properties of Hilbert spaces 27
2.5 Invariant subspaces 32
2.6 Schur decomposition 34
2.7 Non-unitary transformations 41
3 Jacobi iteration 47
3.1 Iterated similarity transformations 47
3.2 Two-dimensional Schur decomposition 48
3.3 One step of the iteration 51
3.4 Error estimates 55
3.5 Quadratic convergence 61
4 Power methods 69
4.1 Power iteration 69
4.2 Rayleigh quotient 74
4.3 Residual-based error control 78
4.4 Inverse iteration 81
4.5 Rayleigh iteration 85
4.6 Convergence to invariant subspace 87
4.7 Simultaneous iteration 91
4.8 Convergence for general matrices 99
5 QR iteration 108
5.1 Basic QR step 108
5.2 Hessenberg form 112
5.3 Shifting 121
5.4 Deflation 124
5.5 Implicit iteration 126
5.6 Multiple-shift strategies 134
6 Bisection methods 140
6.1 Sturm chains 142
6.2 Gershgorin discs 149
7 Krylov subspace methods for large sparse eigenvalue problems 153
7.1 Sparse matrices and projection methods 153
7.2 Krylov subspaces 157
7.3 Gram-Schmidt process 160
7.4 Arnoldi iteration 167
7.5 Symmetric Lanczos algorithm 172
7.6 Chebyshev polynomials 173
7.7 Convergence of Krylov subspace methods 180
8 Generalized and polynomial eigenvalue problems 190
8.1 Polynomial eigenvalue problems and linearization 190
8.2 Matrix pencils 193
8.3 Deflating subspaces and the generalized Schur decomposition 197
8.4 Hessenberg-triangular form 200
8.5 Deflation 204
8.6 The QZ step 206
Bibliography 211
Index 214

lt;P>„Hinreichend ausführliche und gut verständliche Beweise."
Prof. Dr. Alexander Hornberg, Hochschule Esslingen