Introduction to Computational Linear Algebra (Hardcover)

Basic Linear Algebra Subprograms: BLAS
An Introductory Example
Matrix Notations
IEEE Floating Point Systems and Computer Arithmetic
Vector-Vector Operations: Level-1 BLAS
Matrix-Vector Operations: Level-2 BLAS
Matrix-Matrix Operations: Level-3 BLAS
Sparse Matrices: Storage and Associated Operations

Basic Concepts for Matrix Computations
Vector Norms
Complements on Square Matrices
Rectangular Matrices: Ranks and Singular Values
Matrix Norms

Gauss Elimination and LU Decompositions of Matrices
Special Matrices for LU Decomposition
Gauss Transforms
Naive LU Decomposition for a Square Matrix with Principal Minor Property (pmp)
Gauss Reduction with Partial Pivoting: PLU Decompositions
MATLAB Commands Related to the LU Decomposition
Condition Number of a Square Matrix

Orthogonal Factorizations and Linear Least Squares Problems
Formulation of Least Squares Problems: Regression Analysis
Existence of Solutions Using Quadratic Forms
Existence of Solutions through Matrix Pseudo-Inverse
The QR Factorization Theorem
Gram-Schmidt Orthogonalization: Classical, Modified, and Block
Solving the Least Squares Problem with the QR Decomposition
Householder QR with Column Pivoting
MATLAB Implementations

Algorithms for the Eigenvalue Problem
Basic Principles
QR Method for a Non-Symmetric Matrix
Algorithms for Symmetric Matrices
Methods for Large Size Matrices
Singular Value Decomposition

Iterative Methods for Systems of Linear Equations
Stationary Methods
Krylov Methods
Method of Steepest Descent for spd Matrices
Conjugate Gradient Method (CG) for spd Matrices
The Generalized Minimal Residual Method
The Bi-Conjugate Gradient Method
Preconditioning Issues

Sparse Systems to Solve Poisson Differential Equations
Poisson Differential Equations
The Path to Poisson Solvers
Finite Differences for Poisson-Dirichlet Problems
Variational Formulations
One-Dimensional Finite-Element Discretizations

Exercises and Computer Exercises appear at the end of each chapter.