An INTRODUCTION to NUMERICAL METHODS

A MATLAB Approach (2nd Edition)

Preface

1. Introduction

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

6.3 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

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

11.7 PREDICTOR-CORRECTOR 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

13.4 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