9780898713619

Numerical Linear Algebra

by ;
  • ISBN13:

    9780898713619

  • ISBN10:

    0898713617

  • Format: Paperback
  • Copyright: 5/1/1997
  • Publisher: Society for Industrial & Applied
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Summary

This is a concise, insightful introduction to the field of numerical linear algebra. The clarity and eloquence of the presentation make it popular with teachers and students alike. The text aims to expand the reader's view of the field and to present standard material in a novel way. All of the most important topics in the field are covered with a fresh perspective, including iterative methods for systems of equations and eigenvalue problems and the underlying principles of conditioning and stability. Presentation is in the form of 40 lectures, which each focus on one or two central ideas. The unity between topics is emphasized throughout, with no risk of getting lost in details and technicalities. The book breaks with tradition by beginning with the QR factorization - an important and fresh idea for students, and the thread that connects most of the algorithms of numerical linear algebra.

Table of Contents

Preface ix
Acknowledgments xi
I Fundamentals 1(38)
Matrix-Vector Multiplication
3(8)
Orthogonal Vectors and Matrices
11(6)
Norms
17(8)
The Singular Value Decomposition
25(7)
More on the SVD
32(7)
II QR Factorization and Least Squares 39(48)
Projectors
41(7)
QR Factorization
48(8)
Gram-Schmidt Orthogonalization
56(7)
MATLAB
63(6)
Householder Triangularization
69(8)
Least Squares Problems
77(10)
III Conditioning and Stability 87(58)
Conditioning and Condition Numbers
89(8)
Floating Point Arithmetic
97(5)
Stability
102(6)
More on Stability
108(6)
Stability of Householder Triangularization
114(7)
Stability of Back Substitution
121(8)
Conditioning of Least Squares Problems
129(8)
Stability of Least Squares Algorithms
137(8)
IV Systems of Equations 145(34)
Gaussian Elimination
147(8)
Pivoting
155(8)
Stability of Gaussian Elimination
163(9)
Cholesky Factorization
172(7)
V Eigenvalues 179(62)
Eigenvalue Problems
181(9)
Overview of Eigenvalue Algorithms
190(6)
Reduction to Hessenberg or Tridiagonal Form
196(6)
Rayleigh Quotient, Inverse Iteration
202(9)
QR Algorithm without Shifts
211(8)
QR Algorithm with Shifts
219(6)
Other Eigenvalue Algorithms
225(9)
Computing the SVD
234(7)
VI Iterative Methods 241(80)
Overview of Iterative Methods
243(7)
The Arnoldi Iteration
250(7)
How Arnoldi Locates Eigenvalues
257(9)
GMRES
266(10)
The Lanczos Iteration
276(9)
From Lanczos to Gauss Quadrature
285(8)
Conjugate Gradients
293(10)
Biorthogonalization Methods
303(10)
Preconditioning
313(8)
Appendix The Definition of Numerical Analysis 321(8)
Notes 329(14)
Bibliography 343(10)
Index 353

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