An introduction to numerical methods: a MATLAB approach/ Abdelwahab Kharab, Ronald B. Guenther

1 Introduction
1.1 ABOUT MATLAB and MATLAB GUI (Graphical User Interface)
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 METHODS
3.7 MULTIPLE ROOTS AND THE MODIFIED
NEWTON METHOD
3 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 PARTIAL
PIVOTING
4.4 LU DECOMPOSITION
4.4.1 Grout s and Cholesky's methods .
4.4.2 Gaussian elimination method
4.5 ITERATIVE METHODS
4.5.1 Jacobi iterative method
4.5.2 Gauss-Seidel iterative method
4.5.3 Gonvergence
APPLIED PROBLEMS
5 Interpolation
5.1 POLYNOMIAL INTERPOLATION THEORY
5.2 NEWTON'S DIVIDED-DIFFERENCE INTERPOLATING
POLYNOMIAL .
5.3 THE ERROR OF THE INTERPOLATING
POLYNOMIAL
5.4 LAGRANGE INTERPOLATING POLYNOMIAL
APPLIED PROBLEMS
6 Interpolation with Spline Functions
6.1 PIECE WISE LINEAR INTERPOLATION .
6.2 QUADRATIC SPLINE
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 .olden 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 ROMBERC ALGORITHM
10.4 GAUSSIAN QUADRATURE
APPLIED PROBLEMS
11 Numerical Methods for Linear Integral Equations
11.1 INTRODUCTION
11.2 QUADRATURE RULES
11.2.1 Trapezoidal rule
11.2.2 The Gauss-Nystrom method
11.3 THE SUCCESSIVE APPROXIMATION METHOD
11.4 SCHMIDT'S METHOD
11.5 VOLTERRA-TYPE INTEGRAL EQUATIONS
11.5.1 Euler's method
11.5.2 Heun's method
APPLIED PROBLEMS
12 Numerical Methods for Differential Equations
12.1 EULER'S METHOD
12.2 ERROR ANALYSIS
12.3 HIGHER-ORDER TAYLOR SERIES METHODS . .
12.4 RUNCE-KUTTA METHODS
12.5 MULTISTEP METHODS . .
12.6 ADAMS-BASHFORTH METHODS
12.7 PREDICTOR-CORRECTOR METHODS
12.8 ADAMS-MOULTON METHODS
12.9 NUMERICAL STABILITY
12.10 HIGHER-ORDER EQUATIONS AND SYSTEMS
OF DIFFERENTIAL EQUATIONS AND SYSTEMS
12.11 TPLICIT METHODS AND STIFF SYSTEMS
12.12 ASE PLANE ANALYSIS: CHAOTIC DIFFERENTIAL
.QUATIONS
APPLIED PROBLEMS .
13 Boundary-Value Problems
13.1 FINITE-DIFFERENCE METHODS
13.2 SHOOTING METHODS
13.2.1 The nonlinear case
13.2.2 The linear case
APPLIED PROBLEMS
14 Eigenvalues and Eigenvectors
14.1 BASIC THEORY
14.2 THE POWER METHOD
14.3 THE QUADRATIC METHOD
14.4 EIGENVALUES FOR BOUNDARY-VALUE PROBLEMS
14.5 BIFURCATIONS IN DIFFERENTIAL EQUATIONS
APPLIED PROBLEMS
15 Partial Differential Equations
15.1 PARABOLIC EQUATIONS
15.1.1 Explicit methods
15.1.2 Implicit methods
15.2 HYPERBOLIC EQUATIONS
15.3 ELLIPTIC EQUATIONS .
15.4 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
15.4.1 Burger's equation
15.4.2 Reaction-diffusion equation
15.4.3 Porous media equation
15.4.4 Hamilton-Jacobi-Bellman equation
15.5 INTRODUCTION TO FINITE-ELEMENT METHOD . .
15.5.1 Theory
15.5.2 The Finite-Element Method
APPLIED PROBLEMS
Bibliography and References
Appendix
A Calculus Review
A.l Limits and continuity
A. 2 Differentiation
A. 3 Integration
B MATLAB Built-in Functions
C Text MATLAB unctions