An Introduction to Numerical Methods: A Matlab Approach

by Kharab, Abdelwahab; Guenther, Ronald B.

• ISBN13: 9781584882817
• ISBN10: 1584882816
• Format: Hardcover
• Copyright: 2001-11-01
• Publisher: CRC Pr I Llc

Summary

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 Approachfulfills 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.

Author Biography

Abdelwahab Kharab is an Associate Professor of Mathematics at King Fahd University of Petroleum & Minerals in Dhahran, Saudi Arabia Ronald B. Guenther is a Professor of Mathematics at Oregon State University, Corvallis, USA

Table of Contents

Introduction	1	(20)
About the Software Matlab	1	(1)
An Introduction to Matlab	2	(11)
Matrices and matrix computation	2	(4)
Output format	6	(1)
Planar plots	7	(1)
3-D mesh plots	8	(1)
Function files	9	(1)
Defining functions	10	(1)
Relations and loops	11	(2)
Taylor Series	13	(8)
Number System and Errors	21	(16)
Floating-Point Arithmetic	21	(5)
Round-Off Errors	26	(3)
Truncation Error	29	(1)
Interval Arithmetic	30	(7)
Roots of Equations	37	(42)
The Bisection Method	37	(7)
The Method of False Position	44	(4)
The Secant Method	48	(4)
Newton's Method	52	(6)
Convergence of the Newton and Secant Methods	58	(2)
Fixed Point Iteration	60	(6)
Multiple Roots and the Modified Newton Method	66	(6)
Newton's Method for Nonlinear Systems	72	(7)
System of Linear Equations	79	(52)
Matrices and Matrix Operations	79	(4)
Naive Gaussian Elimination	83	(9)
Gaussian Elimination with Scaled Partial Pivoting	92	(10)
Lu Decomposition	102	(13)
Crout's and Choleski's methods	102	(6)
Gaussian elimination method	108	(7)
Iterative Methods	115	(16)
Jacobi iterative method	115	(4)
Gauss-Seidel iterative method	119	(4)
Convergence	123	(8)
Interpolation	131	(22)
Polynomial Interpolation Theory	132	(2)
Newton's Divided Difference Interpolating Polynomial	134	(10)
The Error of the Interpolating Polynomial	144	(3)
Lagrange Interpolating Polynomial	147	(6)
Interpolation with Spline Functions	153	(22)
Piecewise Linear Interpolation	153	(5)
Quadratic Spline	158	(3)
Natural Cubic Splines	161	(14)
The Method of Least Squares	175	(28)
Linear Least Squares	176	(6)
Least Squares Polynomial	182	(8)
Nonlinear Least Squares	190	(7)
Exponential form	191	(2)
Hyperbolic form	193	(4)
Trigonometric Least Squares Polynomial	197	(6)
Numerical Differentiation and Integration	203	(44)
Numerical Differentiation	204	(4)
Richardson's Formula	208	(6)
Trapezoidal Rule	214	(8)
Simpson's Rule	222	(8)
Romberg Algorithm	230	(7)
Gaussian Quadrature	237	(10)
Numerical Methods for Ordinary Differential Equations	247	(64)
Euler's Method	247	(7)
Error Analysis	254	(5)
Higher Order Taylor Series Methods	259	(3)
Runge-Kutta Methods	262	(10)
Multistep Methods	272	(1)
Adams-Bashforth Method	273	(9)
Predictor-Corrector Methods	282	(1)
Adams-Moulton Method	283	(6)
Numerical Stability	289	(3)
Higher Order Equations and Systems	292	(7)
Implicit Methods and Stiff Systems	299	(2)
Phase Plane Analysis: Chaotic Equations	301	(10)
Boundary-Value Problems	311	(18)
Finite-Difference Methods	311	(7)
Shooting Methods	318	(11)
The nonlinear case	318	(3)
The linear case	321	(8)
Eigenvalues and Eigenvectors	329	(30)
Basic Theory	329	(5)
The Power Method	334	(4)
The Quadratic Method	338	(11)
Eigenvalues for Boundary-Value Problems	349	(3)
Bifurcations in Differential Equations	352	(7)
Partial Differential Equations	359	(30)
Parabolic Equations	360	(13)
Explicit methods	360	(6)
Implicit methods	366	(7)
Hyperbolic Equations	373	(7)
Elliptic Equations	380	(9)
Bibliography and References	389	(4)
Appendix A Calculus Review	393	(4)
A.0.1 Limits and continuity	393	(1)
A.0.2 Differentiation	394	(1)
A.0.3 Integration	394	(3)
Appendix B MATLAB Built-In Functions	397	(4)
Appendix C Text MATLAB Functions	401	(2)
Answers to Selected Exercises	403	(24)
Index	427