1st Edition

# Introduction to Computational Linear Algebra

**Also available as eBook on:**

*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.