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

by Kharab; Abdelwahab

• ISBN13: 9781584885573
• ISBN10: 1584885572
• Edition: 2nd
• Format: Hardcover
• Copyright: 2005-07-27
• Publisher: Chapman & Hall/

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 availability of today's computers virtually demand 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.Updated to reflect the latest version of MATLAB, the second edition of An Introduction to Numerical Methods continues to fulfill both these needs. It introduces the theory and applications of the most commonly used techniques for solving numerical problems on a computer. It covers a wide range of useful algorithms, each presented with full details so that readers can visualize and interpret each step.Highlights of the second edition:· A new chapter on numerical optimization· New sections on finite elements· More exercises and applied problems in each chapter· MATLAB incorporated as an integral part of the textEmphasis 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

1 Introduction	1	(24)
1.1 ABOUT THE SOFTWARE MATLAB	1	(1)
1.2 AN INTRODUCTION TO MATLAB	2	(15)
1.2.1 Matrices and matrix computation	2	(5)
1.2.2 Polynomials	7	(1)
1.2.3 Output format	8	(1)
1.2.4 Planar plots	9	(1)
1.2.5 3-D mesh plots	10	(1)
1.2.6 Function files	11	(1)
1.2.7 Defining functions	12	(1)
1.2.8 Relations and loops	13	(4)
1.3 TAYLOR SERIES	17	(8)
2 Number System and Errors	25	(20)
2.1 FLOATING-POINT ARITHMETIC	25	(5)
2.2 ROUND-OFF ERRORS	30	(6)
2.3 TRUNCATION ERROR	36	(2)
2.4 INTERVAL ARITHMETIC	38	(7)
3 Roots of Equations	45	(66)
3.1 THE BISECTION METHOD	47	(8)
3.2 THE METHOD OF FALSE POSITION	55	(7)
3.3 FIXED-POINT ITERATION	62	(8)
3.4 THE SECANT METHOD	70	(6)
3.5 NEWTON'S METHOD	76	(11)
3.6 CONVERGENCE OF THE NEWTON AND SECANT METHODS	87	(3)
3.7 MULTIPLE ROOTS AND THE MODIFIED NEWTON METHOD	90	(7)
3.8 NEWTON'S METHOD FOR NONLINEAR SYSTEMS	97	(6)
APPLIED PROBLEMS	103	(8)
4 System of Linear Equations	111	(68)
4.1 MATRICES AND MATRIX OPERATIONS	112	(4)
4.2 NAIVE GAUSSIAN ELIMINATION	116	(8)
4.3 GAUSSIAN ELIMINATION WITH SCALED PARTIAL PIVOTING	124	(15)
4.4 LU DECOMPOSITION	139	(16)
4.4.1 Croft's and Choleski's methods	140	(4)
4.4.2 Gaussian elimination method	144	(11)
4.5 ITERATIVE METHODS	155	(15)
4.5.1 Jacobi iterative method	156	(3)
4.5.2 Gauss-Seidel iterative method	159	(1)
4.5.3 Convergence	160	(10)
APPLIED PROBLEMS	170	(9)
5 Interpolation	179	(32)
5.1 POLYNOMIAL INTERPOLATION THEORY	180	(3)
5.2 NEWTON'S DIVIDED DIFFERENCE INTERPOLATINGPOLYNOMIAL	183	(12)
5.3 THE ERROR OF THE INTERPOLATING POLYNOMIAL	195	(6)
5.4 LAGRANGE INTERPOLATING POLYNOMIAL	201	(6)
APPLIED PROBLEMS	207	(4)
6 Interpolation with Spline Functions	211	(32)
6.1 PIECEWISE LINEAR INTERPOLATION	212	(7)
6.2 QUADRATIC SPLINE	219	(5)
6.3 NATURAL CUBIC SPLINES	224	(16)
APPLIED PROBLEMS	240	(3)
7 The Method of Least Squares	243	(32)
7.1 LINEAR LEAST SQUARES	244	(7)
7.2 LEAST SQUARES POLYNOMIAL	251	(9)
7.3 NONLINEAR LEAST SQUARES	260	(9)
7.3.1 Exponential form	260	(2)
7.3.2 Hyperbolic form	262	(7)
7.4 TRIGONOMETRIC LEAST SQUARES POLYNOMIAL	269	(3)
APPLIED PROBLEMS	272	(3)
8 Numerical Optimization	275	(26)
8.1 ANALYSIS OF SINGLE-VARIABLE FUNCTIONS	276	(2)
8.2 LINE SEARCH METHODS	278	(16)
8.2.1 Bracketing the minimum	278	(1)
8.2.2 Golden section search	279	(4)
8.2.3 Fibonacci Search	283	(3)
8.2.4 Parabolic Interpolation	286	(8)
8.3 MINIMIZATION USING DERIVATIVES	294	(4)
8.3.1 Newton's method	294	(1)
8.3.2 Secant method	295	(3)
APPLIED PROBLEMS	298	(3)
9 Numerical Differentiation	301	(20)
9.1 NUMERICAL DIFFERENTIATION	301	(8)
9.2 RICHARDSON'S FORMULA	309	(7)
APPLIED PROBLEMS	316	(5)
10 Numerical Integration	321	(50)
10.1 TRAPEZOIDAL RULE	322	(11)
10.2 SIMPSON'S RULE	333	(11)
10.3 ROMBERG ALGORITHM	344	(9)
10.4 GAUSSIAN QUADRATURE	353	(12)
APPLIED PROBLEMS	365	(6)
11 Numerical Methods for Differential Equations	371	(86)
11.1 EULER'S METHOD	372	(8)
11.2 ERROR ANALYSIS	380	(5)
11.3 HIGHER ORDER TAYLOR SERIES METHODS	385	(5)
11.4 RUNGE-KUTTA METHODS	390	(16)
11.5 MULTISTEP METHODS	406	(1)
11.6 ADAMS-BASHFORTH METHODS	406	(11)
11.7 PREDICTOR-CORRECTOR METHODS	417	(1)
11.8 ADAMS-MOULTON METHODS	418	(9)
11.9 NUMERICAL STABILITY	427	(4)
11.10 HIGHER ORDER EQUATIONS AND SYSTEMS OF DIFFERENTIAL EQUATIONS	431	(7)
11.11 IMPLICIT METHODS AND STIFF SYSTEMS	438	(3)
11.12 PHASE PLANE ANALYSIS: CHAOTIC DIFFERENTIAL EQUATIONS	441	(6)
APPLIED PROBLEMS	447	(10)
12 Boundary-Value Problems	457	(28)
12.1 FINITE-DIFFERENCE METHODS	458	(9)
12.2 SHOOTING METHODS	467	(13)
12.2.1 The nonlinear case	467	(5)
12.2.2 The linear case	472	(8)
APPLIED PROBLEMS	480	(5)
13 Eigenvalues and Eigenvectors	485	(30)
13.1 BASIC THEORY	485	(5)
13.2 THE POWER METHOD	490	(4)
13.3 THE QUADRATIC METHOD	494	(11)
13.4 EIGENVALUES FOR BOUNDARY-VALUE PROBLEMS	505	(3)
13.5 BIFURCATIONS IN DIFFERENTIAL EQUATIONS	508	(5)
APPLIED PROBLEMS	513	(2)
14 Partial Differential Equations	515	(44)
14.1 PARABOLIC EQUATIONS	516	(13)
14.1.1 Explicit methods	516	(5)
14.1.2 Implicit methods	521	(8)
14.2 HYPERBOLIC EQUATIONS	529	(7)
14.3 ELLIPTIC EQUATIONS	536	(7)
14.4 INTRODUCTION TO FINITE-ELEMENT METHOD	543	(14)
14.4.1 Theory	543	(8)
14.4.2 The Finite Element Method	551	(6)
APPLIED PROBLEMS	557	(2)
Bibliography and References	559	(6)
Appendix	565	(10)
A Calculus Review	565	(4)
A.1 Limits and continuity	565	(1)
A.2 Differentiation	566	(1)
A.3 Integration	567	(2)
B MATLAB Built-in Functions	569	(4)
C Text MATLAB Functions	573	(2)
Answers to Selected Exercises	575	(28)
Index	603