Numerical Linear Algebra

by Trefethen, Lloyd N.; Bau, David

• ISBN13: 9780898713619
• ISBN10: 0898713617
• Format: Paperback
• Copyright: 1997-05-01
• Publisher: SOCIETY FOR INDUSTRIAL & APPLIED MATHEMATICS

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