# An Introduction to Numerical Methods

## A MATLAB Approach, Third Edition

© 2011 – Chapman and Hall/CRC

576 pages | 92 B/W Illus.

Hardback: 9781439868997
pub: 2011-11-16
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eBook (VitalSource) : 9781439889145
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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.

### Reviews

Praise for Previous Editions

Kharab and Guenther offer an attractive, clear, error-free, and well-written introduction to numerical methods … Highly recommended.

—J.H. Ellison, CHOICE

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

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

Applied problems

Numerical Methods for Linear Integral Equations

Introduction

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

Predictor-corrector 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

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

Index

#### Abdelwahab Kharab

Abu Dhabi, , UAE

Abdelwahab Kharab is an associate professor in the College of Arts and Sciences at Abu Dhabi University. His research interests include numerical analysis and simulation for the numerical solution of partial differential equations that arise in science and engineering.

Ronald B. Guenther is an Emeritus Professor in the Department of Mathematics at Oregon State University. His research interests include mathematically modeling deterministic systems and the ordinary and partial differential equations that arise from these models.