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E-grāmata: Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms

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  • Formāts: 464 pages
  • Izdošanas datums: 01-Apr-2012
  • Izdevniecība: Princeton University Press
  • Valoda: eng
  • ISBN-13: 9781400842674
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Numerical Methods: Design, Analysis, and Computer Implementation of AlgorithmsPalielināt
  • Formāts: 464 pages
  • Izdošanas datums: 01-Apr-2012
  • Izdevniecība: Princeton University Press
  • Valoda: eng
  • ISBN-13: 9781400842674
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Numerical Methods provides a clear and concise exploration of standard numerical analysis topics, as well as nontraditional ones, including mathematical modeling, Monte Carlo methods, Markov chains, and fractals. Filled with appealing examples that will motivate students, the textbook considers modern application areas, such as information retrieval and animation, and classical topics from physics and engineering. Exercises use MATLAB and promote understanding of computational results.

The book gives instructors the flexibility to emphasize different aspects--design, analysis, or computer implementation--of numerical algorithms, depending on the background and interests of students. Designed for upper-division undergraduates in mathematics or computer science classes, the textbook assumes that students have prior knowledge of linear algebra and calculus, although these topics are reviewed in the text. Short discussions of the history of numerical methods are interspersed throughout the chapters. The book also includes polynomial interpolation at Chebyshev points, use of the MATLAB package Chebfun, and a section on the fast Fourier transform. Supplementary materials are available online.

  • Clear and concise exposition of standard numerical analysis topics
  • Explores nontraditional topics, such as mathematical modeling and Monte Carlo methods
  • Covers modern applications, including information retrieval and animation, and classical applications from physics and engineering
  • Promotes understanding of computational results through MATLAB exercises
  • Provides flexibility so instructors can emphasize mathematical or applied/computational aspects of numerical methods or a combination
  • Includes recent results on polynomial interpolation at Chebyshev points and use of the MATLAB package Chebfun
  • Short discussions of the history of numerical methods interspersed throughout
  • Supplementary materials available online

Recenzijas

"Distinguishing features are the inclusion of many recent applications of numerical methods and the extensive discussion of methods based on Chebyshev interpolation. This book would be suitable for use in courses aimed at advanced undergraduate students in mathematics, the sciences, and engineering."--Choice "An instructor could assemble several different one-semester courses using this book--numerical linear algebra and interpolation, or numerical solutions of differential equations--or perhaps a two-semester sequence. This is a charming book, well worth consideration for the next numerical analysis course."--William J. Satzer, MAA Focus

Preface xiii
1 Mathematical Modeling
1(18)
1.1 Modeling in Computer Animation
2(2)
1.1.1 A Model Robe
2(2)
1.2 Modeling in Physics: Radiation Transport
4(2)
1.3 Modeling in Sports
6(2)
1.4 Ecological Models
8(3)
1.5 Modeling a Web Surfer and Google
11(3)
1.5.1 The Vector Space Model
11(2)
1.5.2 Google's PageRank
13(1)
1.6
Chapter 1 Exercises
14(5)
2 Basic Operations with MATLAB
19(22)
2.1 Launching MATLAB
19(1)
2.2 Vectors
20(2)
2.3 Getting Help
22(1)
2.4 Matrices
23(1)
2.5 Creating and Running .m Files
24(1)
2.6 Comments
25(1)
2.7 Plotting
25(2)
2.8 Creating Your Own Functions
27(1)
2.9 Printing
28(1)
2.10 More Loops and Conditionals
29(2)
2.11 Clearing Variables
31(1)
2.12 Logging Your Session
31(1)
2.13 More Advanced Commands
31(1)
2.14
Chapter 2 Exercises
32(9)
3 Monte Carlo Methods
41(30)
3.1 A Mathematical Game of Cards
41(5)
3.1.1 The Odds in Texas Holdem
42(4)
3.2 Basic Statistics
46(10)
3.2.1 Discrete Random Variables
48(3)
3.2.2 Continuous Random Variables
51(2)
3.2.3 The Central Limit Theorem
53(3)
3.3 Monte Carlo Integration
56(8)
3.3.1 Buffon's Needle
56(2)
3.3.2 Estimating π
58(2)
3.3.3 Another Example of Monte Carlo Integration
60(4)
3.4 Monte Carlo Simulation of Web Surfing
64(3)
3.5
Chapter 3 Exercises
67(4)
4 Solution of a Single Nonlinear Equation in One Unknown
71(36)
4.1 Bisection
75(5)
4.2 Taylor's Theorem
80(3)
4.3 Newton's Method
83(6)
4.4 Quasi-Newton Methods
89(4)
4.4.1 Avoiding Derivatives
89(1)
4.4.2 Constant Slope Method
89(1)
4.4.3 Secant Method
90(3)
4.5 Analysis of Fixed Point Methods
93(5)
4.6 Fractals, Julia Sets, and Mandelbrot Sets
98(4)
4.7
Chapter 4 Exercises
102(5)
5 Floating-Point Arithmetic
107(17)
5.1 Costly Disasters Caused by Rounding Errors
108(2)
5.2 Binary Representation and Base 2 Arithmetic
110(2)
5.3 Floating-Point Representation
112(2)
5.4 IEEE Floating-Point Arithmetic
114(2)
5.5 Rounding
116(2)
5.6 Correctly Rounded Floating-Point Operations
118(1)
5.7 Exceptions
119(1)
5.8
Chapter 5 Exercises
120(4)
6 Conditioning of Problems: Stability of Algorithms
124(7)
6.1 Conditioning of Problems
125(1)
6.2 Stability of Algorithms
126(3)
6.3
Chapter 6 Exercises
129(2)
7 Direct Methods for Solving Linear Systems and Least Squares Problems
131(50)
7.1 Review of Matrix Multiplication
132(1)
7.2 Gaussian Elimination
133(18)
7.2.1 Operation Counts
137(2)
7.2.2 LU Factorization
139(2)
7.2.3 Pivoting
141(3)
7.2.4 Banded Matrices and Matrices for Which Pivoting Is Not Required
144(4)
7.2.5 Implementation Considerations for High Performance
148(3)
7.3 Other Methods for Solving Ax = b
151(3)
7.4 Conditioning of Linear Systems
154(10)
7.4.1 Norms
154(4)
7.4.2 Sensitivity of Solutions of Linear Systems
158(6)
7.5 Stability of Gaussian Elimination with Partial Pivoting
164(2)
7.6 Least Squares Problems
166(9)
7.6.1 The Normal Equations
167(1)
7.6.2 QR Decomposition
168(3)
7.6.3 Fitting Polynomials to Data
171(4)
7.7
Chapter 7 Exercises
175(6)
8 Polynomial and Piecewise Polynomial Interpolation
181(31)
8.1 The Vandermonde System
181(1)
8.2 The Lagrange Form of the Interpolation Polynomial
181(4)
8.3 The Newton Form of the Interpolation Polynomial
185(5)
8.3.1 Divided Differences
187(3)
8.4 The Error in Polynomial Interpolation
190(2)
8.5 Interpolation at Chebyshev Points and chebfun
192(5)
8.6 Piecewise Polynomial Interpolation
197(7)
8.6.1 Piecewise Cubic Hermite Interpolation
200(1)
8.6.2 Cubic Spline Interpolation
201(3)
8.7 Some Applications
204(2)
8.8
Chapter 8 Exercises
206(6)
9 Numerical Differentiation and Richardson Extrapolation
212(15)
9.1 Numerical Differentiation
213(8)
9.2 Richardson Extrapolation
221(4)
9.3
Chapter 9 Exercises
225(2)
10 Numerical Integration
227(24)
10.1 Newton--Cotes Formulas
227(5)
10.2 Formulas Based on Piecewise Polynomial Interpolation
232(2)
10.3 Gauss Quadrature
234(6)
10.3.1 Orthogonal Polynomials
236(4)
10.4 Clenshaw--Curtis Quadrature
240(2)
10.5 Romberg Integration
242(1)
10.6 Periodic Functions and the Euler-Maclaurin Formula
243(4)
10.7 Singularities
247(1)
10.8
Chapter 10 Exercises
248(3)
11 Numerical Solution of the Initial Value Problem for Ordinary Differential Equations
251(49)
11.1 Existence and Uniqueness of Solutions
253(4)
11.2 One-Step Methods
257(18)
11.2.1 Euler's Method
257(5)
11.2.2 Higher-Order Methods Based on Taylor Series
262(1)
11.2.3 Midpoint Method
262(2)
11.2.4 Methods Based on Quadrature Formulas
264(1)
11.2.5 Classical Fourth-Order Runge--Kutta and Runge--Kutta--Fehlberg Methods
265(2)
11.2.6 An Example Using MATLAB's ODE Solver
267(3)
11.2.7 Analysis of One-Step Methods
270(2)
11.2.8 Practical Implementation Considerations
272(2)
11.2.9 Systems of Equations
274(1)
11.3 Multistep Methods
275(9)
11.3.1 Adams--Bashforth and Adams--Moulton Methods
275(2)
11.3.2 General Linear m-Step Methods
277(3)
11.3.3 Linear Difference Equations
280(3)
11.3.4 The Dahlquist Equivalence Theorem
283(1)
11.4 Stiff Equations
284(7)
11.4.1 Absolute Stability
285(4)
11.4.2 Backward Differentiation Formulas (BDF Methods)
289(1)
11.4.3 Implicit Runge--Kutta (IRK) Methods
290(1)
11.5 Solving Systems of Nonlinear Equations in Implicit Methods
291(4)
11.5.1 Fixed Point Iteration
292(1)
11.5.2 Newton's Method
293(2)
11.6
Chapter 11 Exercises
295(5)
12 More Numerical Linear Algebra: Eigenvalues and Iterative Methods for Solving Linear Systems
300(50)
12.1 Eigenvalue Problems
300(27)
12.1.1 The Power Method for Computing the Largest Eigenpair
310(3)
12.1.2 Inverse Iteration
313(2)
12.1.3 Rayleigh Quotient Iteration
315(1)
12.1.4 The QR Algorithm
316(4)
12.1.5 Google's PageRank
320(7)
12.2 Iterative Methods for Solving Linear Systems
327(18)
12.2.1 Basic Iterative Methods for Solving Linear Systems
327(1)
12.2.2 Simple Iteration
328(4)
12.2.3 Analysis of Convergence
332(4)
12.2.4 The Conjugate Gradient Algorithm
336
12.2.5 Methods for Nonsymmetric Linear Systems
334(11)
12.3
Chapter 12 Exercises
345(5)
13 Numerical Solution of Two-Point Boundary Value Problems
350(29)
13.1 An Application: Steady-State Temperature Distribution
350(2)
13.2 Finite Difference Methods
352(13)
13.2.1 Accuracy
354(6)
13.2.2 More General Equations and Boundary Conditions
360(5)
13.3 Finite Element Methods
365(9)
13.3.1 Accuracy
372(2)
13.4 Spectral Methods
374(2)
13.5
Chapter 13 Exercises
376(3)
14 Numerical Solution of Partial Differential Equations
379(42)
14.1 Elliptic Equations
381(7)
14.1.1 Finite Difference Methods
381(5)
14.1.2 Finite Element Methods
386(2)
14.2 Parabolic Equations
388(8)
14.2.1 Semidiscretization and the Method of Lines
389(1)
14.2.2 Discretization in Time
389(7)
14.3 Separation of Variables
396(6)
14.3.1 Separation of Variables for Difference Equations
400(2)
14.4 Hyperbolic Equations
402(7)
14.4.1 Characteristics
402(1)
14.4.2 Systems of Hyperbolic Equations
403(1)
14.4.3 Boundary Conditions
404(1)
14.4.4 Finite Difference Methods
404(5)
14.5 Fast Methods for Poisson's Equation
409(5)
14.5.1 The Fast Fourier Transform
411(3)
14.6 Multigrid Methods
414(4)
14.7
Chapter 14 Exercises
418(3)
Appendix A Review of Linear Algebra
421(15)
A.1 Vectors and Vector Spaces
421(1)
A.2 Linear Independence and Dependence
422(1)
A.3 Span of a Set of Vectors; Bases and Coordinates; Dimension of a Vector Space
423(1)
A.4 The Dot Product; Orthogonal and Orthonormal Sets; the Gram--Schmidt Algorithm
423(2)
A.5 Matrices and Linear Equations
425(2)
A.6 Existence and Uniqueness of Solutions; the Inverse; Conditions for Invertibility
427(4)
A.7 Linear Transformations; the Matrix of a Linear Transformation
431(1)
A.8 Similarity Transformations; Eigenvalues and Eigenvectors
432(4)
Appendix B Taylor's Theorem in Multidimnsions
436(3)
References 439(6)
Index 445
Anne Greenbaum is professor of applied mathematics at the University of Washington. She is the author of Iterative Methods for Solving Linear Systems. Timothy P. Chartier is associate professor of mathematics at Davidson College.
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