9780133198492

Applied Numerical Analysis Using Matlab

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  • ISBN13:

    9780133198492

  • ISBN10:

    0133198499

  • Format: Paperback
  • Copyright: 2008-01-01
  • Publisher: PRENTICE HALL
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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

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