An introduction to numerical methods: a MATLAB approach/
Abdelwahab Kharab, Ronald B. Guenther
- 3rd ed.
- Boca Raton: CRC Press, 2012.
- 567 p. : ill. ; 26 cm. + CD-ROM
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