Explorations in Numerical Analysis


James V. Lambers, The University of Southern Mississippi
Amber C. Sumner, William Carey University

Published by World Scientific (buy here)
Also available at Amazon


This textbook introduces advanced undergraduate and early-career graduate students to the field of numerical analysis. This field pertains to the design, analysis, and implementation of algorithms for the approximate solution of mathematical problems that arise in applications spanning science and engineering, and are not practical to solve using analytical techniques such as those taught in courses in calculus, linear algebra or differential equations. Topics covered include error analysis, computer arithmetic, solution of systems of linear equations, least squares problems, eigenvalue problems, polynomial interpolation and approximation, numerical differentiation and integration, nonlinear equations, optimization, ordinary differential equations, and partial differential equations. For each problem considered, the presentation includes the derivation of solution techniques, analysis of their efficiency, accuracy and robustness, and details of their implementation, illustrated through the MATLAB programming language. This text is suitable for a year-long sequence in numerical analysis, and can also be used for a one-semester course in numerical linear algebra.

Table of Contents

  1. Preliminaries
    1. What is Numerical Analysis?
    2. Understanding Error
  2. Numerical Linear Algebra
    1. Direct Methods for Linear Systems
    2. Least Squares Problems
    3. Iterative Methods for Linear Systems
    4. Eigenvalue Problems
  3. Data Fitting and Function Approximation
    1. Polynomial Interpolation
    2. Approximation of Functions
    3. Differentiation and Integration
  4. Nonlinear Equations and Optimization
    1. Zeros of Nonlinear Functions
    2. Optimization
  5. Differential Equations
    1. Initial Value Problems
    2. Two-Point Boundary Value Problems
    3. Partial Differential Equations
    1. Review of Calculus
    2. Review of Linear Algebra