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 ComputationsVector 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.

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.