Author(s): Lloyd N. Trefethen, David Bau III
Publisher: Society for Industrial and Applied Mathematics
Year: 1997
Language: English
Commentary: bookmarked
Pages: 376
City: Philadelphia
Title Page......Page 1
Contents......Page 9
Preface......Page 11
Acknowledgments......Page 13
I Fundamentals......Page 15
Lecture 1 Matrix-Vector Multiplication......Page 17
Lecture 2 Orthogonal Vectors and Matrices......Page 25
Lecture 3 Norms......Page 31
Lecture 4 The Singular Value Decomposition......Page 39
Lecture 5 More on the SVD......Page 46
II QR Factorization and Least Squares......Page 53
Lecture 6 Projectors......Page 55
Lecture 7 QR Factorization......Page 62
Lecture 8 Gram-Schmidt Orthogonalization......Page 70
Lecture 9 MATLAB......Page 77
Lecture 10 Householder Triangularization......Page 83
Lecture 11 Least Squares Problems......Page 91
III Conditioning and Stability......Page 101
Lecture 12 Conditioning and Condition Numbers......Page 103
Lecture 13 Floating Point Arithmetic......Page 111
Lecture 14 Stability......Page 116
Lecture 15 More on Stability......Page 122
Lecture 16 Stability of Householder Triangularization......Page 128
Lecture 17 Stability of Back Substitution......Page 135
Lecture 18 Conditioning of Least Squares Problems......Page 143
Lecture 19 Stability of Least Squares Algorithms......Page 151
IV Systems of Equations......Page 159
Lecture 20 Gaussian Elimination......Page 161
Lecture 21 Pivoting......Page 169
Lecture 22 Stability of Gaussian Elimination......Page 177
Lecture 23 Cholesky Factorization......Page 186
V Eigenvalues......Page 193
Lecture 24 Eigenvalue Problems......Page 195
Lecture 25 Overview of Eigenvalue Algorithms......Page 204
Lecture 26 Reduction to Hessenberg or Tridiagonal Form......Page 210
Lecture 27 Rayleigh Quotient, Inverse Iteration......Page 216
Lecture 28 QR Algorithm without Shifts......Page 225
Lecture 29 QR Algorithm with Shifts......Page 233
Lecture 30 Other Eigenvalue Algorithms......Page 239
Lecture 31 Computing the SVD......Page 248
VI Iterative Methods......Page 255
Lecture 32 Overview of Iterative Methods......Page 257
Lecture 33 The Arnoldi Iteration......Page 264
Lecture 34 How Arnoldi Locates Eigenvalues......Page 271
Lecture 35 GMRES......Page 280
Lecture 36 The Lanczos Iteration......Page 290
Lecture 37 From Lanczos to Gauss Quadrature......Page 299
Lecture 38 Conjugate Gradients......Page 307
Lecture 39 Biorthogonalization Methods......Page 317
Lecture 40 Preconditioning......Page 327
Appendix The Definition of Numerical Analysis......Page 335
Notes......Page 343
Bibliography......Page 357
Index......Page 367
