1st Edition

Introduction to Computational Linear Algebra

    259 Pages 9 B/W Illustrations
    by Chapman & Hall

    Continue Shopping

    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



    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.