Explorations in Numerical Analysis

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 and ordinary 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.

Contents: Preliminaries:What is Numerical Analysis?Understanding ErrorNumerical Linear Algebra:Direct Methods for Linear SystemsLeast Squares ProblemsIterative Methods for Linear SystemsEigenvalue ProblemsData Fitting and Function Approximation:Polynomial InterpolationApproximation of FunctionsDifferentiation and IntegrationNonlinear Equations and Optimization:Zeros of Nonlinear FunctionsOptimizationDifferential Equations:Initial Value ProblemsTwo-Point Boundary Value ProblemsPartial Differential EquationsAppendices:Review of CalculusReview of Linear Algebra
Readership: Advanced undergraduate and graduate students in numerical analysis, approximations and expansions, linear and multilinear algebra/matrix theory and ordinary differential equations.
Key Features:The goal of this textbook is to provide the reader an experience that is more conducive to arousing interest in numerical analysis, thereby enhancing understanding. This is to be accomplished by giving the reader a more hands-on experience with the material than existing textbooks through MATLAB exercisesThe purpose of these exercises is to augment the text with visualization and experimentation in order to involve the reader in the derivation and analysis of each algorithm, as well as its implementationThe intent of this book is to aid the reader to understand the text through a combination of exposition about the evolution of problem-solving techniques and the theoretical results that support them, and coding exercises and demonstrations to illustrate this evolution