An INTRODUCTION to NUMERICAL METHODS

A MATLAB Approach (2nd Edition)

 

Table of Contents

 

 

Preface

 

1. Introduction

 1.1 ABOUT THE SOFTWARE MATLAB

 1.2 AN INTRODUCTION TO MATLAB

  1.2.1 Matrices and matrix computation

  1.2.2 Polynomials

  1.2.3 Output format

  1.2.4 Planar plots

  1.2.5 3-D mesh plots

  1.2.6 Function files

  1.2.7 Defining functions

  1.2.8 Relations and loops

 1.3 TAYLOR SERIES

 

2. Number System and Errors

 2.1 FLOATING-POINT ARITHMETIC

 2.2 ROUND-OFF ERRORS

 2.3 TRUNCATION ERROR

 2.4 INTERVAL ARITHMETIC

 

3. Roots of Equations

 3.1 THE BISECTION METHOD

 3.2 THE METHOD OF FALSE POSITION

 3.3 FIXED-POINT ITERATION  

 3.4 THE SECANT METHOD

 3.5 NEWTON'S METHOD

 3.6 CONVERGENCE OF THE NEWTON AND SECANT

 3.7 MULTIPLE ROOTS AND THE MODIFIED NEWTON

 3.8 NEWTON'S METHOD FOR NONLINEAR SYSTEMS

 APPLIED PROBLEMS

 

4. System of Linear Equations

 4.1 MATRICES AND MATRIX OPERATIONS

 4.2 NAIVE GAUSSIAN ELIMINATION

 4.3 GAUSSIAN ELIMINATION WITH SCALED PIVOTING

 4.4 LU DECOMPOSITION

  4.4.1 Crout's and Choleski's methods

  4.4.2 Gaussian elimination method1

 4.5 ITERATIVE METHODS

  4.5.1 Jacobi iterative method

  4.5.2 Gauss-Seidel iterative method

  4.5.3 Convergence

 APPLIED PROBLEMS

 

5. Interpolation

 5.1 POLYNOMIAL INTERPOLATION THEORY

 5.2 NEWTON'S DIVIDED DIFFERENCE POLYNOMIAL

 5.3 THE ERROR OF THE INTERPOLATING POLYNOMIAL

 5.4 LAGRANGE INTERPOLATING POLYNOMIAL

 APPLIED PROBLEMS

 

6. Interpolation with Spline Functions

 6.1 PIECEWISE LINEAR INTERPOLATION

 6.2 QUADRATIC SPLINES

 6.3 NATURAL CUBIC SPLINES

 APPLIED PROBLEMS

 

7. The Method of Least Squares

 7.1 LINEAR LEAST SQUARES

 7.2 LEAST SQUARES POLYNOMIAL

 7.3 NONLINEAR LEAST SQUARES

  7.3.1 Exponential form

  7.3.2 Hyperbolic form

 7.4 TRIGONOMETRIC LEAST SQUARES POLYNOMIAL

 APPLIED PROBLEMS

 

8. Numerical Optimization

 8.1 ANALYSIS OF SINGLE-VARIABLE FUNCTIONS

 8.2 LINE SEARCH METHODS

  8.2.1 Bracketing the minimum

  8.2.2 Golden section search

  8.2.3 Fibonacci Search

  8.2.4 Parabolic Interpolation

 8.3 MINIMIZATION USING DERIVATIVES

  8.3.1 Newton's method

  8.3.2 Secant method

APPLIED PROBLEMS

 

9. Numerical Differentiation

 9.1 NUMERICAL DIFFERENTIATION

 9.2 RICHARDSON'S FORMULA

APPLIED PROBLEMS

 

10. Numerical Integration

 10.1 TRAPEZOIDAL RULE

 10.2 SIMPSON'S RULE

 10.3 ROMBERG ALGORITHM

 10.4 GAUSSIAN QUADRATURE

APPLIED PROBLEMS

 

11. Numerical Methods for Differential Equations

 11.1 EULER'S METHOD

 11.2 ERROR ANALYSIS

 11.3 HIGHER ORDER TAYLOR SERIES METHODS

 11.4 RUNGE-KUTTA METHODS

 11.5 MULTISTEP METHODS

 11.6 ADAMS-BASHFORTH METHODS

 11.7 PREDICTOR-CORRECTOR METHODS

 11.8 ADAMS-MOULTON METHODS

 11.9 NUMERICAL STABILITY

 11.10 HIGHER ORDER EQUATIONS AND SYSTEMS

 11.11 IMPLICIT METHODS AND STIFF SYSTEMS

 11.12 PHASE PLANE ANALYSIS: CHAOTIC DIFFERENTIAL EQUATIONS

APPLIED PROBLEMS

 

12. Boundary-Value Problems

 12.1 FINITE-DIFFERENCE METHODS

 12.2 SHOOTING METHODS

  12.2.1 The nonlinear case

  12.2.2 The linear case

APPLIED PROBLEMS

 

13. Eigenvalues and Eigenvectors

 13.1 BASIC THEORY

 13.2 THE POWER METHOD

 13.3 THE QUADRATIC METHOD

 13.4 EIGENVALUES FOR BOUNDARY-VALUE PROBLEMS

 13.5 BIFURCATIONS IN DIFFERENTIAL EQUATIONS

APPLIED PROBLEMS

 

14. Partial Differential Equations

 14.1 PARABOLIC EQUATIONS

  14.1.1 Explicit methods

  14.1.2 Implicit methods

 14.2 HYPERBOLIC EQUATIONS

 14.3 ELLIPTIC EQUATIONS

 14.4 INTRODUCTION TO THE FINITE ELEMENT METHOD

  14.4.1 Theory

  14.4.2 The finite Element Method

APPLIED PROBLEMS

 

Appendix

 A Calculus Review

  A.0.1 Limits and continuity

  A.0.2 Differentiation

  A.0.3 Integration

 B MATLAB Built-in Functions

 C Text MATLAB Functions

 

Answers to Selected Exercises

 

Index

 

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