1st Edition
Introduction to Computational Linear Algebra
Teach Your Students Both the Mathematics of Numerical Methods and the Art of Computer Programming
Introduction to Computational Linear Algebra presents classroom-tested material on computational linear algebra and its application to numerical solutions of partial and ordinary differential equations. The book is designed for senior undergraduate students in mathematics and engineering as well as first-year graduate students in engineering and computational science.
The text first introduces BLAS operations of types 1, 2, and 3 adapted to a scientific computer environment, specifically MATLAB®. It next covers the basic mathematical tools needed in numerical linear algebra and discusses classical material on Gauss decompositions as well as LU and Cholesky’s factorizations of matrices. The text then shows how to solve linear least squares problems, provides a detailed numerical treatment of the algebraic eigenvalue problem, and discusses (indirect) iterative methods to solve a system of linear equations. The final chapter illustrates how to solve discretized sparse systems of linear equations. Each chapter ends with exercises and computer projects.
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
Bibliography
Index
Exercises and Computer Exercises appear at the end of each chapter.
Biography
Nabil Nassif is affiliated with the Department of Mathematics at the American University of Beirut, where he teaches and conducts research in mathematical modeling, numerical analysis, and scientific computing. He earned a PhD in applied mathematics from Harvard University under the supervision of Professor Garrett Birkhoff.
Jocelyne Erhel is a senior research scientist and scientific leader of the Sage team at INRIA in Rennes, France. She earned a PhD from the University of Paris. Her research interests include sparse linear algebra and high performance scientific computing applied to geophysics, mainly groundwater models.
Bernard Philippe was a senior research scientist at INRIA in Rennes, France, until 2015 when he retired. He earned a PhD from the University of Rennes. His research interests include matrix computing with a special emphasis on large-sized eigenvalue problems.
