An Introduction to Numerical Methods: A MATLAB Approach

Numerical methods are a mainstay of researchers and professionals across the many mathematics, scientific, and engineering disciplines. The importance of these methods combined with the power and widespread availability of desktop computers virtually demands that students in these fields be well-versed not only in the numerical techniques, but also in the use of a modern computational software package. An Introduction to Numerical Methods: A MATLAB Approach fulfills both these needs. Requiring only calculus and some basic programming concepts as background, it introduces readers to the theory and applications of the most commonly used techniques for solving numerical problems on a personal computer. It covers a wide range of useful algorithms, each presented with full details so that readers can visualize and interpret each step. The enclosed CD-ROM contains simple MATLAB functions that provide step-by-step explanations of the mechanisms behind the algorithm for each technique and guide readers through the necessary calculations. Emphasis on understanding how the methods work, a simple, direct style, and thorough coverage make this book an outstanding initiation that allows students to see almost immediate results. It will boost their confidence in their ability to master the subject and give them valuable experience in the use of MATLAB.

Table of Contents:
Introduction About the software MATLAB An Introduction to MATLAB Taylor Series Number System and Errors Floating-Point Arithmetic Round-Off Errors Truncation Error Interval Arithmetic Roots of Equations The Bisection Method The Method of False Position The Secant method Newton’s Method Convergence of the Newton and Secant Methods Fixed Point Iteration Multiple Roots and the Modified Newton Method Newton’s Method for Nonlinear Systems System of Linear Equations Matrices and Matrix Operations Naïve Gaussian Elimination Gaussian Elimination with Scaled Partial Pivoting Lu Decomposition Iterative Methods Interpolation Polynomial Interpolation Theory Newton’s Divided Difference Interpolating Polynomial The Error of the Interpolating Polynomial Lagrange Interpolating Polynomial Interpolation with Spline Functions Piecewise Linear Interpolation Quadratic Spline Natural Cubic Splines The Method of Least Squares Linear Least Squares Least Squares Polynomial Nonlinear Least Squares Trigonometric Least Squares Polynomial Numerical Differentiation and Integration Numerical Differentiation Richardson’s Formula Trapezoidal Rule Simpson’s Rule Romberg Algorithm Gaussian Quadrature Numerical Methods for Ordinary Differential Equations Euler’s Method Error Analysis Higher Order Taylor Series Methods Runge-Kutta Methods Multistep Methods Adams-Bashforth Method Predictor-Corrector Methods Adams-Moulton Method Numerical Stability Higher Order Equations and Systems of Equations Implicit Methods and Stiff Systems Phase Plane Analysis: Chaotic Equations Boundary Value Problems Finite-Difference Methods Shooting Methods Eigenvalues and Eigenvectors Basic Theory The Power Method The Quadratic Method Eigenvalues for Boundary-Value Problems Bifurcations in Differential Equations Partial Differential Equations Parabolic Equations Hyperbolic Equations Elliptic Equations Bibliography and References Appendices Calculus Review MATLAB built-in functions Text MATLAB functions Answers to Selected Exercises Index