An Introduction to Numerical Methods: A MATLAB Approach, Third Edition

Highly recommended by CHOICE, previous editions of this popular textbook offered an accessible and practical introduction to numerical analysis. An Introduction to Numerical Methods: A MATLAB® Approach, Third Edition continues to present a wide range of useful and important algorithms for scientific and engineering applications. The authors use MATLAB to illustrate each numerical method, providing full details of the computer results so that the main steps are easily visualized and interpreted. New to the Third Edition A chapter on the numerical solution of integral equations A section on nonlinear partial differential equations (PDEs) in the last chapter Inclusion of MATLAB GUIs throughout the text The book begins with simple theoretical and computational topics, including computer floating point arithmetic, errors, interval arithmetic, and the root of equations. After presenting direct and iterative methods for solving systems of linear equations, the authors discuss interpolation, spline functions, concepts of least-squares data fitting, and numerical optimization. They then focus on numerical differentiation and efficient integration techniques as well as a variety of numerical techniques for solving linear integral equations, ordinary differential equations, and boundary-value problems. The book concludes with numerical techniques for computing the eigenvalues and eigenvectors of a matrix and for solving PDEs. CD-ROM Resource The accompanying CD-ROM contains simple MATLAB functions that help students understand how the methods work. These functions provide a clear, step-by-step explanation of the mechanism behind the algorithm of each numerical method and guide students through the calculations necessary to understand the algorithm. Written in an easy-to-follow, simple style, this text improves students’ ability to master the theoretical and practical elements of the methods. Through this book, they will be able to solve many numerical problems using MATLAB.

Table of Contents:
Introduction About MATLAB and MATLAB graphical user interface (GUI) 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 Fixed-point iteration The secant method Newton’s method Convergence of the Newton and Secant methods Multiple roots and the modified Newton method Newton’s method for nonlinear systems Applied problems System of Linear Equations Matrices and matrix operations Naïve Gaussian elimination Gaussian elimination with scaled partial pivoting Lu decomposition Iterative methods Applied problems Interpolation Polynomial interpolation theory Newton’s divided-difference interpolating polynomial The error of the interpolating polynomial Lagrange interpolating polynomial Applied problems Interpolation with Spline Functions Piecewise linear interpolation Quadratic spline Natural cubic splines Applied problems The Method of Least Squares Linear least squares Least-squares polynomial Nonlinear least squares Trigonometric least-squares polynomial Applied problems Numerical Optimization Analysis of single-variable functions Line search methods Minimization using derivatives Applied problems Numerical Differentiation Numerical differentiation Richardson’s formula Applied problems Numerical Integration Trapezoidal rule Simpson’s rule Romberg algorithm Gaussian quadrature Applied problems Numerical Methods for Linear Integral Equations Introduction Quadrature rules The successive approximation method Schmidt’s method Volterra-type integral equations Applied problems Numerical Methods for Differential Equations Euler’s Method Error Analysis Higher-order Taylor series methods Runge-Kutta methods Adams-Bashforth methods Predictor-corrector methods Adams-Moulton methods Numerical stability Higher-order equations and systems of differential equations Implicit methods and stiff systems Phase plane analysis: chaotic differential equations Applied problems Boundary-Value Problems Finite-difference methods Shooting methods Applied problems Eigenvalues and Eigenvectors Basic theory The power method The quadratic method Eigenvalues for boundary-value problems Bifurcations in differential equations Applied problems Partial Differential Equations Parabolic equations Hyperbolic equations Elliptic equations Nonlinear partial differential equations Introduction to finite-element method Applied problems Bibliography and References Appendix A: Calculus Review Appendix B: MATLAB Built-in Functions Appendix C: Text MATLAB Functions Appendix D: MATLAB GUI Answers to Selected Exercises Index