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This book brings together linear algebra, numerical methods and an easy to use programming environment under Matlab (or Scilab). One of the key features of the book are the worked out examples and exercises at the end of each chapter. The reader is asked to do some numerical experiments in Matlab and then to prove the results theoretically. The book is a combination and update of two earlier French books by the authors. It is appropriate for both undergraduate and beginning graduate courses in mathematics as well as for working scientists and engineers as a self-study tool and reference. This book is about numerical linear algebra and focuses on practical algorithms for solving computer problems of linear algebra.
Discretization of a Differential Equation
Least Squares Fitting
Vibrations of a Mechanical System
The Vibrating String
Image Compression by the SVD Factorization
Definition and Properties of Matrices
Gram–Schmidt Orthonormalization Process
Matrices
Spectral Theory of Matrices
Matrix Triangularization
Matrix Diagonalization
Min–Max Principle
Singular Values of a Matrix
Matrix Norms, Sequences, and Series
Matrix Norms and Subordinate Norms
Subordinate Norms for Rectangular Matrices
Matrix Sequences and Series
Introduction to Algorithmics
Algorithms and pseudolanguage
Operation Count and Complexity
The Strassen Algorithm
Equivalence of Operations
LinearSystems
Square Linear Systems
Over- and Underdetermined Linear Systems
Numerical Solution
Direct Methods for Linear Systems
Gaussian Elimination Method
LU Decomposition Method
Cholesky Method
QR Factorization Method
Least Squares Problems
Motivation
Main Results
Numerical Algorithms
Simple Iterative Methods
General Setting
Jacobi, Gauss–Seidel, and Relaxation Methods
The Special Case of Tridiagonal Matrices
Discrete Laplacian
Programming Iterative Methods
Block Methods
Conjugate Gradient Method
The Gradient Method
Geometric Interpretation
Some Ideas for Further Generalizations
Theoretical Definition of the Conjugate Gradient Method
Conjugate Gradient Algorithm
Methods for Computing Eigenvalues
Generalities
Conditioning
Power Method
Jacobi Method
Givens–Householder Method
QR Method
Lanczos Method
Solutions and Programs
Index of Programs