The Lanczos method: evolution and application

Cover......Page 1
S Title......Page 2
Editors......Page 3
Title......Page 4
ISBN 0-89871-537-7......Page 5
Dedications......Page 6
Contents......Page 8
Preface......Page 12
Part I EVOLUTION......Page 14
1.1 The eigenvalue problem......Page 16
1.2 The method of minimized iterations......Page 17
1.3 Calculation of eigenvalues and eigenvectors......Page 19
1.4 Geometric interpretation......Page 21
2.1 Computational formulation......Page 22
2.2 Solving the tridiagonal problem......Page 24
2.3 Exact arithmetic algorithm......Page 25
2.4 Computational example......Page 26
3.1 Breakdown of the Lanczos process......Page 30
3.2 Measuring and maintaining orthogonality......Page 31
3.3 Monitoring convergence and estimating accuracy......Page 32
3.4 Finite precision algorithm......Page 33
4.1 The block Lanczos method......Page 36
4.2 Measuring and maintaining block orthogonality......Page 38
4.4 Block real symmetric algorithm......Page 39
5.1 Block biorthogonal Lanczos method......Page 42
5.3 Error analysis......Page 43
Algorithm 5.4.1. Block unsymmetric algorithm......Page 44
5.5 Adapting the block size......Page 45
5.6 Preventing breakdown......Page 46
5.7 Maintaining biorthonormality......Page 47
Part II APPLICATIONS......Page 50
6.1 Spectral transformat......Page 52
6.2 Frequency domain decomposition......Page 54
6.3 Geometric domain decomposition......Page 55
6.3.2 Partitioned matrix decomposition......Page 56
6.3.3 Partitioned substitution......Page 57
6.4 Parallel computational strategy......Page 58
7.1 Analysis of mechanical systems......Page 60
7.2 Generalized linear eigenvalue problem......Page 62
7.3 Normal modes analysis application......Page 63
8.1 Generalized quadratic eigenvalue problem......Page 66
8.2 Recovery of physical solution......Page 67
8.3 Orthogonality diagnostics......Page 69
8.4 Implicit operator multiplication......Page 70
Algorithm 8.5.2. Transpose implicit operator algorithm......Page 72
8.6 Complex eigenvalue analysis applicati......Page 73
9.1 The interior acoustic problem......Page 76
9.2 Fluid-structure interaction......Page 78
9.3 Coupled forced vibration problem......Page 79
9.4.1 Transfer function calculat......Page 80
9.4.2 Approximate transfer function......Page 81
9.5 Acoustic optimization application......Page 82
10.1 Exact solution......Page 84
10.2 Approximate solution......Page 85
10.3 Recursive approximate solution......Page 86
Algorithm 10.4.1. Lanczos linear solver algorithm......Page 87
10.5 Linear static analysis application......Page 88
Closing Remarks......Page 90
A Brief Biography of Cornelius Lanczos......Page 92
Bibliography......Page 94
Index......Page 98