Applied Numerical Analysis Using Matlab

by Fausett, Laurene V.

• ISBN13: 9780133198492
• ISBN10: 0133198499
• Format: Paperback
• Copyright: 2008-01-01
• Publisher: PRENTICE HALL

Summary

Each chapter uses introductory problems from specific applications. These easy-to-understand problems clarify for the reader the need for a particular mathematical technique. Numerical techniques are explained with an emphasis on why they work. FEATURES bull; bull;Discussion of the contexts and reasons for selection of each problem and solution method. bull;Worked-out examples are very realistic and not contrived. bull;MATLAB code provides an easy test-bed for algorithmic ideas.

Table of Contents

PREFACE	XI
1 FOUNDATIONS	1	(40)
1.1 Applied Problems	3	(3)
1.1.1 Nonlinear Functions	3	(1)
1.1.2 Linear Systems	4	(1)
1.1.3 Numerical Integration	5	(1)
1.2 Numerical Techniques	6	(3)
1.2.1 Fixed-Point Iteration	6	(1)
1.2.2 Gaussian Elimination	7	(1)
1.2.3 Trapezoid Rule	8	(1)
1.3 Analysis	9	(16)
1.3.1 Key Issues for Iterative Methods	10	(4)
1.3.2 How Good Is the Result?	14	(7)
1.3.3 Getting Better Results	21	(4)
1.4 Using MATLAB	25	(16)
1.4.1 Command Window Computation	26	(5)
1.4.2 MATLAB Programs	31	(10)
2 SOLVING EQUATIONS OF ONE VARIABLE	41	(36)
2.1 Bisection Method	44	(4)
2.1.1 MATLAB Function for Bisection	47	(1)
2.1.2 Discussion	48	(1)
2.2 Regula Falsi and Secant Methods	48	(10)
2.2.1 Regula Falsi	49	(5)
2.2.2 Secant Method	54	(4)
2.3 Newton's Method	58	(5)
2.3.1 MATLAB Function for Newton's Method	60	(1)
2.3.2 Discussion	61	(2)
2.4 Muller's Method	63	(6)
2.4.1 MATLAB Function for Muller's Method	65	(2)
2.4.2 Discussion	67	(2)
2.5 Fixed-Point Iteration	69	(1)
2.6 MATLAB's Methods	69	(8)
3 SOLVING SYSTEMS OF LINEAR EQUATIONS: DIRECT METHODS	77	(32)
3.1 Gaussian Elimination	80	(8)
3.1.1 Using Matrix Notation	81	(3)
3.1.2 MATLAB Function for Basic Gaussian Elimination	84	(1)
3.1.3 Discussion	85	(3)
3.2 Gaussian Elimination with Row Pivoting	88	(5)
3.2.1 MATLAB Function for Gaussian Elimination with Row Pivoting	90	(2)
3.2.2 Discussion	92	(1)
3.3 Gaussian Elimination for Tridiagonal Systems	93	(6)
3.3.1 MATLAB Function for Solving a Tridiagonal System	97	(1)
3.3.2 Discussion	97	(2)
3.4 MATLAB'S Methods	99	(10)
4 SOLVING SYSTEMS OF LINEAR EQUATIONS: ITERATIVE METHODS	109	(26)
4.1 Jacobi Method	111	(7)
4.1.1 MATLAB Function for Jacobi Method	114	(2)
4.1.2 Discussion	116	(2)
4.2 Gauss-Seidel Method	118	(5)
4.2.1 MATLAB Function for Gauss-Seidel Method	120	(2)
4.2.2 Discussion	122	(1)
4.3 Successive Over Relaxation	123	(4)
4.3.1 MATLAB Function for SOR	124	(1)
4.3.2 Discussion	125	(1)
4.3.3 Some Useful Theoretical Results	126	(1)
4.4 MATLAB'S Methods	127	(8)
5 NONLINEAR FUNCTIONS OF SEVERAL VARIABLES	135	(24)
5.1 Newton's Method for Nonlinear Systems	138	(7)
5.1.1 Matrix-Vector Notation	140	(2)
5.1.2 MATLAB Function for Newton's Method for Nonlinear Systems	142	(3)
5.2 Fixed-Point Iteration for Nonlinear Systems	145	(4)
5.2.1 MATLAB Function for Fixed-Point Iteration	146	(2)
5.2.2 Discussion	148	(1)
5.3 Minimum of a Nonlinear Function of Several Variables	149	(4)
5.3.1 MATLAB Function for Minimization by Gradient Descent	151	(2)
5.4 MATLAB Methods	153	(6)
6 LU FACTORIZATION	159	(30)
6.1 LU Factorization from Gaussian Elimination	161	(4)
6.1.1 MATLAB Function for LU Factorization	163	(1)
6.1.2 Discussion	164	(1)
6.2 LU Factorization of Tridiagonal Matrices	165	(2)
6.2.1 MATLAB Function for LU Factorization of a Tridiagonal Matrix	165	(2)
6.3 LU Factorization with Pivoting	167	(4)
6.3.1 MATLAB Function for LU Factorization with Row Pivoting	168	(1)
6.3.2 Discussion	169	(2)
6.4 Direct LU Factorization	171	(4)
6.4.1 Doolittle LU Factorization	171	(2)
6.4.2 Cholesky LU Factorization	173	(2)
6.5 Application of LU Factorization	175	(6)
6.5.1 Solving Systems of Linear Equations	175	(2)
6.5.2 Solving a Tridiagonal System Using LU Factorization	177	(1)
6.5.3 Determinant of a Matrix	178	(1)
6.5.4 Inverse of a Matrix	179	(2)
6.6 MATLAB'S Methods	181	(8)
7 EIGENVALUES, EIGENVECTORS, AND QR FACTORIZATION	189	(38)
7.1 Power Method	192	(10)
7.1.1 Basic Power Method	193	(4)
7.1.2 Shifted Power Method	197	(1)
7.1.3 Inverse Power Method	198	(1)
7.1.4 General Inverse Power Method	200	(1)
7.1.5 Discussion	201	(1)
7.2 QR Factorization	202	(7)
7.2.1 Householder and Givens Transformations	202	(6)
7.2.2 Basic QR Factorization	208	(1)
7.3 Finding Eigenvalues Using QR Factorization	209	(7)
7.3.1 Basic QR Eigenvalue Method	210	(2)
7.3.2 Improved QR Eigenvalue Method	212	(4)
7.4 MATLAB'S Methods	216	(11)
8 INTERPOLATION	227	(60)
8.1 Polynomial Interpolation	230	(20)
8.1.1 Lagrange Interpolation Polynomials	230	(8)
8.1.2 Newton Interpolation Polynomials	238	(8)
8.1.3 Difficulties with Polynomial Interpolation	246	(4)
8.2 Hermite Interpolation	250	(4)
8.3 Rational-Function Interpolation	254	(3)
8.4 Spline Interpolation	257	(11)
8.4.1 Piecewise Linear Interpolation	258	(1)
8.4.2 Piecewise Quadratic Interpolation	258	(3)
8.4.3 Piecewise Cubic Interpolation	261	(7)
8.5 MATLAB'S Interpolation Functions	268	(19)
8.5.1 Interpolation in One Dimension	268	(1)
8.5.2 Interpolation in Two Dimensions	269	(18)
9 FUNCTION APPROXIMATION	287	(42)
9.1 Least Squares Approximation	290	(17)
9.1.1 Linear Least Squares Approximation	290	(6)
9.1.2 Quadratic Least Squares Approximation	296	(5)
9.1.3 Cubic Least Squares Approximation	301	(3)
9.1.4 Least Squares Approximation for Other Functional Forms	304	(3)
9.2 Continuous Least Squares	307	(8)
9.2.1 Continuous Least Squares with Orthogonal Polynomials	310	(1)
9.2.2 Gram-Schmidt Process	311	(1)
9.2.3 Legendre Polynomials	312	(1)
9.2.4 Least Squares Approximation with Legendre Polynomials	313	(2)
9.3 Function Approximation at a Point	315	(4)
9.3.1 Taylor Approximation	316	(1)
9.3.2 Pade Approximation	316	(3)
9.4 Using MATLAB'S Functions	319	(10)
10 FOURIER METHODS	329	(36)
10.1 Fourier Approximation and Interpolation	332	(10)
10.1.1 MATLAB Function for Fourier Interpolation or Approximation	335	(2)
10.1.2 Discussion	337	(5)
10.2 Fast Fourier Transforms for n = 2(r)	342	(10)
10.2.1 Discrete Fourier Transform	342	(1)
10.2.2 Fast Fourier Transform	343	(1)
10.2.3 Matrix Form of FFT	344	(2)
10.2.4 Algebraic Form of FFT	346	(2)
10.2.5 MATLAB Function for FFT with n = 4	348	(4)
10.3 Fast Fourier Transforms for General n	352	(3)
10.3.1 MATLAB Function for FFT with n = rs	354	(1)
10.4 Using MATLAB'S Functions	355	(10)
11 NUMERICAL DIFFERENTIATION AND INTEGRATION	365	(38)
11.1 Differentiation	368	(8)
11.1.1 First Derivatives	368	(4)
11.1.2 Higher Derivatives	372	(2)
11.1.3 Richardson Extrapolation	374	(2)
11.2 Basic Numerical Integration	376	(8)
11.2.1 Trapezoid Rule	377	(2)
11.2.2 Simpson Rule	379	(3)
11.2.3 Midpoint Rule	382	(1)
11.2.4 Other Newton-Cotes Open Formulas	383	(1)
11.3 Better Numerical Integration	384	(8)
11.3.1 Composite Trapezoid Rule	384	(2)
11.3.2 Composite Simpson's Rule	386	(2)
11.3.3 Extrapolation Methods for Quadrature	388	(4)
11.4 Gaussian Quadrature	392	(4)
11.4.1 Gaussian Quadrature on [-1, 1]	392	(1)
11.4.2 Gaussian Quadrature on [a, b]	393	(3)
11.5 MATLAB'S Methods	396	(7)
11.5.1 Differentiation	396	(1)
11.5.2 Integration	396	(7)
12 ORDINARY DIFFERENTIAL EQUATIONS: INITIAL-VALUE PROBLEMS	403	(38)
12.1 Taylor Methods	405	(6)
12.1.1 Euler's Method	405	(4)
12.1.2 Higher Order Taylor Methods	409	(2)
12.2 Runge-Kutta Methods	411	(12)
12.2.1 Midpoint Method	411	(3)
12.2.2 Other Second-Order Runge-Kutta Methods	414	(2)
12.2.3 Third-Order Runge-Kutta Methods	416	(1)
12.2.4 Classic Runge-Kutta Method	417	(3)
12.2.5 Other Runge--Kutta Methods	420	(2)
12.2.6 Runge--Kutta-Fehlberg Methods	422	(1)
12.3 Multistep Methods	423	(8)
12.3.1 Adams-Bashforth Methods	424	(2)
12.3.2 Adams-Moulton Methods	426	(1)
12.3.3 Predictor-Corrector Methods	427	(4)
12.4 Stability	431	(3)
12.5 MATLAB'S Methods	434	(7)
13 SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS	441	(36)
13.1 Higher Order ODEs	444	(1)
13.2 Systems of Two First-Order ODEs	445	(6)
13.2.1 Euler's Method for Solving Two ODE-IVPs	445	(3)
13.2.2 Midpoint Method for Solving Two ODE-IVPs	448	(3)
13.3 Systems of First-Order ODE-IVPs	451	(14)
13.3.1 Euler's Method for Solving Systems of ODEs	452	(1)
13.3.2 Runge-Kutta Methods for Solving Systems of ODEs	453	(6)
13.3.3 Multistep Methods for Systems	459	(6)
13.4 Stiff ODE and Ill-Conditioned Problems	465	(2)
13.5 Using MATLAB'S Functions	467	(10)
14 ORDINARY DIFFERENTIAL EQUATIONS: BOUNDARY VALUE PROBLEMS	477	(26)
14.1 Shooting Method for Solving Linear BVPs	480	(6)
14.1.1 Simple Boundary Conditions	480	(4)
14.1.2 General Boundary Condition at x = b	484	(1)
14.1.3 General Boundary Conditions at Both Ends of the Interval	485	(1)
14.2 Shooting Method for Solving Nonlinear BVPs	486	(6)
14.2.1 Nonlinear Shooting Based on the Secant Method	486	(2)
14.2.2 Nonlinear Shooting Using Newton's Method	488	(4)
14.3 Finite-Difference Method for Solving Linear BVPs	492	(5)
14.4 Finite-Difference Method for Solving Nonlinear BVPs	497	(6)
15 PARTIAL DIFFERENTIAL EQUATIONS	503	(54)
15.1 Heat Equation: Parabolic PDE	507	(15)
15.1.1 Explicit Method for Solving Heat Equation	509	(5)
15.1.2 Implicit Method for Solving Heat Equation	514	(4)
15.1.3 Crank-Nicolson Method for Solving Heat Equation	518	(3)
15.1.4 Heat Equation with Insulated Boundary	521	(1)
15.2 Wave Equation: Hyperbolic PDE	522	(6)
15.2.1 Explicit Method for Solving Wave Equation	523	(4)
15.2.2 Implicit Method for Solving Wave Equation	527	(1)
15.3 Poisson Equation: Elliptic PDE	528	(6)
15.4 Finite-Element Method for Solving an Elliptic PDE	534	(12)
15.5 Using MATLAB'S Functions	546	(11)
BIBLIOGRAPHY	557	(6)
ANSWERS TO SELECTED EXERCISES	563	(20)
SUBJECT INDEX	583	(10)
AUTHOR INDEX	593