Numerical Analysis with CD-ROM / Edition 1

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Overview

Numerical Analysis, designed to be used in a one-year course in engineering, science and mathematics, helps the readers gain a deeper understanding of numerical analysis by highlighting the five major ideas of the discipline: Convergence, Complexity, Conditioning, Compression, and Orthogonality and connecting back to them throughout the text. Each chapter contains a Reality Check, an extended foray into a relevant application area that can be used as a springboard for individual or team projects. MATLAB is used throughout to demonstrate and implement numerical methods.

Fundamentals. Solving Equations. Systems of Equations. Interpolation. Least Square. Numerical Differentiation and Integration. Ordinary Differential Equations. Boundary Value Problems. Partial Differential Equations. Random Numbers and Applications. Trigonometric Interpolation and the FFT. Compression. Eigenvalues and Singular Values. Optimization.

For all readers interested in numerical analysis.

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Product Details

  • ISBN-13: 9780321268983
  • Publisher: Pearson
  • Publication date: 12/8/2005
  • Edition description: BK&CD-ROM
  • Edition number: 1
  • Pages: 688
  • Product dimensions: 7.62 (w) x 9.42 (h) x 1.11 (d)

Meet the Author

Timothy Sauer earned the Ph.D. degree in mathematics at the University of California, Berkeley in 1982, and is currently a professor at George Mason University. He has published articles on a wide range of topics in applied mathematics, including dynamical systems, computational mathematics, and mathematical biology.
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Table of Contents

CHAPTER 0. Fundamentals

0.1 Evaluating a polynomial

0.2 Binary numbers

0.2.1 Decimal to binary

0.2.2 Binary to decimal

0.3 Floating point representation of real numbers

0.3.1 Floating point formats

0.3.2 Machine representation

0.3.3 Addition of floating point numbers

0.4 Loss of significance

0.5 Review of calculus

0.6 Software and Further Reading

CHAPTER 1. Solving Equations

1.1 The Bisection Method

1.1.1 Bracketing a root

1.1.2 How accurate and how fast?

1.2 Fixed point iteration

1.2.1 Fixed points of a function

1.2.2 Geometry of Fixed Point Iteration

1.2.3 Linear Convergence of Fixed Point Iteration

1.2.4 Stopping criteria

1.3 Limits of accuracy

1.3.1 Forward and backward error

1.3.2 The Wilkinson polynomial

1.3.3 Sensitivity and error magnification

1.4 Newton’s Method

1.4.1 Quadratic convergence of Newton’s method

1.4.2 Linear convergence of Newton’s method

1.5 Root-finding without derivatives

1.5.1 Secant method and variants

1.5.2 Brent’s Method

REALITY CHECK 1: Kinematics of the Stewart platform

1.6 Software and Further Reading

CHAPTER 2. Systems of Equations

2.1 Gaussian elimination

2.1.1 Naive Gaussian elimination

2.1.2 Operation counts

2.2 The LU factorization

2.2.1 Backsolving with the LU factorization

2.2.2 Complexity of the LU factorization

2.3 Sources of error

2.3.1 Error magnification and condition number

2.3.2 Swamping

2.4 The PA=LU factorization

2.4.1 Partial pivoting

2.4.2 Permutation matrices

2.4.3 PA = LU factorization

2.4.4 Matlab commands for linear systems

2.5 Iterative methods

2.5.1 Jacobi Method

2.5.2 Gauss-Seidel Method and SOR

2.5.3 Convergence of iterative methods

2.5.4 Sparse matrix computations

REALITY CHECK 2: The Euler-Bernoulli Beam

2.6 Conjugate Gradient Method

2.6.1 Positive-definite matrices

2.6.2 Conjugate Gradient Method

2.7 Nonlinear systems of equations

2.7.1 Multivariate Newton’s method

2.7.2 Broyden’s method

2.8 Software and Further Reading

CHAPTER 3. Interpolation

3.1 Data and interpolating functions

3.1.1 Lagrange interpolation

3.1.2 Newton’s divided differences

3.1.3 How many degree d polynomials pass through n points?

3.1.4 Code for interpolation

3.1.5 Representing functions by approximating polynomials

3.2 Interpolation error

3.2.1 Interpolation error formula

3.2.2 Proof of Newton form and error formula

3.2.3 Runge phenomenon

3.3 Chebyshev interpolation

3.3.1 Chebyshev’s Theorem

3.3.2 Chebyshev polynomials

3.3.3 Change of interval

3.4 Cubic splines

3.4.1 Properties of splines

3.4.2 Endpoint conditions

3.5 B´ezier curves

REALITY CHECK 3: Constructing fonts from B´ezier splines

3.6 Software and Further Reading

CHAPTER 4. Least Squares

4.1 Inconsistent systems of equations

4.2 Linear and nonlinear models

4.2.1 Periodic data

4.2.2 Data linearization

4.3 QR factorization

4.3.1 Gram-Schmidt orthogonalization and least squares

4.3.2 Householder reflectors

4.4 Nonlinear least squares

4.4.1 Gauss-Newton method

4.4.2 Models with nonlinear coefficients

REALITY CHECK 4: GPS, conditioning and nonlinear least squares

4.5 Software and Further Reading

CHAPTER 5. Numerical Differentiation and Integration

5.1 Numerical differentiation

5.1.1 Finite difference formulas

5.1.2 Rounding error

5.1.3 Extrapolation

5.1.4 Symbolic differentiation and integration

5.2 Newton-Cotes formulas for numerical integration

5.2.1 Three simple integrals for Newton-Cotes Formulas

5.2.2 Trapezoid rule

5.2.3 Simpson’s Rule

5.2.4 Composite Newton-Cotes Formulas

5.2.5 Open Newton-Cotes methods

5.3 Romberg integration

5.4 Adaptive quadrature

5.5 Gaussian quadrature

REALITY CHECK 5: Motion control in computer-aided modelling

5.6 Software and Further Reading

CHAPTER 6. Ordinary Differential Equations

6.1 Initial value problems

6.1.1 Euler’s method

6.1.2 Existence, uniqueness, and continuity for solutions

6.1.3 First-order linear equations

6.2 Analysis of IVP solvers

6.2.1 Local and global truncation error

6.2.2 The explicit trapezoid method

6.2.3 Taylor methods

6.3 Systems of ordinary differential equations

6.3.1 Higher order equations

6.3.2 The pendulum

6.3.3 Orbital mechanics

6.4 Runge-Kutta methods and applications

6.4.1 Classical examples

REALITY CHECK 6: The Tacoma Narrows Bridge

6.5 Variable step-size methods

6.6 Implicit methods and stiff equations

6.7 Multistep methods

6.7.1 Generating multistep methods

6.7.2 Explicit multistep methods

6.7.3 Implicit multistep methods

6.8 Software and Further Reading

CHAPTER 7. Boundary Value Problems

7.1 Solutions of boundary value problems

7.1.1 Shooting method

REALITY CHECK 7: Buckling of a circular ring

7.2 Finite difference methods

7.2.1 Linear boundary value problems

7.2.2 Nonlinear boundary value problems

7.3 Collocation and the Finite Element Method

7.3.1 Collocation

7.3.2 Finite elements and the Galerkin method

7.4 Software and Further Reading

CHAPTER 8. Partial Differential Equations

8.1 Parabolic equations

8.1.1 Forward difference method

8.1.2 Stability analysis of forward difference method

8.1.3 Backward difference method

8.1.4 Crank-Nicolson method

8.2 Hyperbolic equations

8.3 Elliptic equations

8.3.1 Finite difference method for elliptic equations

REALITY CHECK 8: Heat distribution on a cooling fin

8.3.2 Finite element method for elliptic equations

8.4 Software and Further Reading

CHAPTER 9. Random Numbers and Applications

9.1 Random numbers

9.1.1 Pseudo-random numbers

9.2 Monte-Carlo simulation

9.2.1 Power laws for Monte Carlo estimation

9.2.2 Quasi-random numbers

9.3 Discrete and continuous Brownian motion

9.3.1 Random walks

9.3.2 Continuous Brownian motion

9.4 Stochastic differential equations

9.4.1 Adding noise to ODEs

9.4.2 Numerical methods for SDEs

REALITY CHECK 9: The Black-Scholes formula

9.5 Software and Further Reading

CHAPTER 10. Trigonometric Interpolation and the FFT

10.1 The Fourier Transform

10.1.1 Complex arithmetic

10.1.2 Discrete Fourier Transform

10.1.3 The Fast Fourier Transform

10.2 Trigonometric interpolation

10.2.1 The DFT Interpolation Theorem

10.2.2 Orthogonality and interpolation

10.2.3 Least squares fitting with trigonometric functions

10.2.4 Sound, noise, and filtering

REALITY CHECK 10: The Wiener filter

10.3 Software and Further Reading

CHAPTER 11. Compression

11.1 The Discrete Cosine Transform

11.1.1 One-dimensional DCT

11.2 Two-dimensional DCT and image compression

11.2.1 The two-dimensional Discrete Cosine Transform

11.2.2 Image compression

11.2.3 Quantization

11.3 Huffman coding

11.3.1 Information theory and coding

11.3.2 Huffman coding for the JPEG format

11.4 Modified DCT and sound compression

11.4.1 Modified Discrete Cosine Transform

11.4.2 Bit quantization

REALITY CHECK 11: A simple audio codec using the MDCT

11.5 Software and Further Reading

CHAPTER 12. Eigenvalues and Singular Values

12.1 Power iteration methods

12.1.1 Power iteration

12.1.2 Convergence of power iteration

12.1.3 Inverse power iteration

12.1.4 Rayleigh quotient iteration

12.2 QR algorithm

12.2.1 Simultaneous iteration

12.2.2 Real Schur form and QR

12.2.3 Householder reflectors

12.2.4 Upper Hessenberg form

REALITY CHECK 12: How search engines rate page quality

12.3 Singular value decomposition

12.3.1 Finding the SVD in general

12.3.2 Special case: symmetric matrices

12.4 Applications of the SVD

12.4.1 Properties of the SVD

12.4.2 Dimension reduction

12.4.3 Compression

12.4.4 Calculating the SVD

12.5 Software and Further Reading

CHAPTER 13. Optimization

13.1 Unconstrained optimization without derivatives

13.1.1 Golden section search

13.1.2 Successive parabolic interpolation

13.1.3 Nelder-Mead search

13.2 Unconstrained optimization with derivatives

13.2.1 Newton’s method

13.2.2 Steepest descent

13.2.3 Conjugate gradient search

13.2.4 Nonlinear least squares

REALITY CHECK 13: Molecular conformation and numerical optimization

13.3 Software and Further Reading

APPENDIX

Appendix A: Matrix Algebra

A.1 Matrix fundamentals

A.2 Block multiplication

A.3 Eigenvalues and eigenvectors

A.4 Symmetric matrices

A.5 Vector calculus

Appendix B: Introduction to Matlab

B.1 Starting Matlab

B.2 Matlab graphics

B.3 Programming in Matlab

B.4 Flow control

B.5 Functions

B.6 Matrix operations

B.7 Animation

Answers to Selected Exercises

Index

Bibliography

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  • Anonymous

    Posted December 20, 2005

    good introduction to numerical analysis

    Sauer teaches numerical analysis with an emphasis on you using Matlab to learn the concepts. His text is suitable as a first course in this subject, at an undergraduate level. The topics are, by and large, unchanged from those in texts written in the 1970s or even earlier. Like the solving of systems of linear equations. A crucial topic which you should master, because these arise throughout engineering and the physical sciences. So the text goes into the ideas of LU factorisation and Gaussian row reduction with pivoting. Other topics covered included numerical integration and differentiation, and least squares curve fitting. The rigour of the proofs is also well chosen. Enough to satisfy most students. But without being too abstruse to put most off. A strong point of the text is the numerous problems interspersed throughout each chapter. These are usefully divided into two types. The first might be considered traditional problems. Where you solve these by hand, with perhaps only a calculator to plug in a few numbers. The second group consists of using Matlab in a more extended foray into numerical analysis. Here, a mere calculator will not suffice. Doing these problems will improve your facility with Matlab and also hopefully garner a general experience in knowing when to use a maths package. It should also be said that if you already use another package, like Mathematica, then this book can still be germane. Roughly speaking, Mathematica, Matlab and Maple have equivalent functionality. Certainly, this is true at the introductory level of the text.

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