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Index
Half title
Title
Copyright
Dedication
Contents
Preface
1 Mathematical Modeling
1.1 Modeling in Computer Animation
1.1.1 A Model Robe
1.2 Modeling in Physics: Radiation Transport
1.3 Modeling in Sports
1.4 Ecological Models
1.5 Modeling a Web Surfer and Google
1.5.1 The Vector Space Model
1.5.2 Google’s PageRank
1.6 Chapter 1 Exercises
2 Basic Operations with Matlab
2.1 Launching MATLAB
2.2 Vectors
2.3 Getting Help
2.4 Matrices
2.5 Creating and Running .m Files
2.6 Comments
2.7 Plotting
2.8 Creating Your Own Functions
2.9 Printing
2.10 More Loops and Conditionals
2.11 Clearing Variables
2.12 Logging Your Session
2.13 More Advanced Commands
2.14 Chapter 2 Exercises
3 Monte Carlo Methods
3.1 A Mathematical Game of Cards
3.1.1 The Odds in Texas Holdem
3.2 Basic Statistics
3.2.1 Discrete Random Variables
3.2.2 Continuous Random Variables
3.2.3 The Central Limit Theorem
3.3 Monte Carlo Integration
3.3.1 Buffon’s Needle
3.3.2 Estimating π
3.3.3 Another Example of Monte Carlo Integration
3.4 Monte Carlo Simulation of Web Surfing
3.5 Chapter 3 Exercises
4 Solution of a Single Nonlinear Equation in One Unknown
4.1 Bisection
4.2 Taylor’s Theorem
4.3 Newton’s Method
4.4 Quasi-Newton Methods
4.4.1 Avoiding Derivatives
4.4.2 Constant Slope Method
4.4.3 Secant Method
4.5 Analysis of Fixed Point Methods
4.6 Fractals, Julia Sets, and Mandelbrot Sets
4.7 Chapter 4 Exercises
5 Floating-Point Arithmetic
5.1 Costly Disasters Caused by Rounding Errors
5.2 Binary Representation and Base 2 Arithmetic
5.3 Floating-Point Representation
5.4 IEEE Floating-Point Arithmetic
5.5 Rounding
5.6 Correctly Rounded Floating-Point Operations
5.7 Exceptions
5.8 Chapter 5 Exercises
6 Conditioning of Problems; Stability of Algorithms
6.1 Conditioning of Problems
6.2 Stability of Algorithms
6.3 Chapter 6 Exercises
7 Direct Methods for Solving Linear Systems and Least Squares Problems
7.1 Review of Matrix Multiplication
7.2 Gaussian Elimination
7.2.1 Operation Counts
7.2.2 LU Factorization
7.2.3 Pivoting
7.2.4 Banded Matrices and Matrices for Which Pivoting Is Not Required
7.2.5 Implementation Considerations for High Performance
7.3 Other Methods for Solving Ax = b
7.4 Conditioning of Linear Systems
7.4.1 Norms
7.4.2 Sensitivity of Solutions of Linear Systems
7.5 Stability of Gaussian Elimination with Partial Pivoting
7.6 Least Squares Problems
7.6.1 The Normal Equations
7.6.2 QR Decomposition
7.6.3 Fitting Polynomials to Data
7.7 Chapter 7 Exercises
8 Polynomial and Piecewise Polynomial Interpolation
8.1 The Vandermonde System
8.2 The Lagrange Form of the Interpolation Polynomial
8.3 The Newton Form of the Interpolation Polynomial
8.3.1 Divided Differences
8.4 The Error in Polynomial Interpolation
8.5 Interpolation at Chebyshev Points and chebfun
8.6 Piecewise Polynomial Interpolation
8.6.1 Piecewise Cubic Hermite Interpolation
8.6.2 Cubic Spline Interpolation
8.7 Some Applications
8.8 Chapter 8 Exercises
9 Numerical Differentiation and Richardson Extrapolation
9.1 Numerical Differentiation
9.2 Richardson Extrapolation
9.3 Chapter 9 Exercises
10 Numerical Integration
10.1 Newton–Cotes Formulas
10.2 Formulas Based on Piecewise Polynomial Interpolation
10.3 Gauss Quadrature
10.3.1 Orthogonal Polynomials
10.4 Clenshaw–Curtis Quadrature
10.5 Romberg Integration
10.6 Periodic Functions and the Euler–Maclaurin Formula
10.7 Singularities
10.8 Chapter 10 Exercises
11 Numerical Solution of the Initial Value Problem for Ordinary Differential Equations
11.1 Existence and Uniqueness of Solutions
11.2 One-Step Methods
11.2.1 Euler’s Method
11.2.2 Higher-Order Methods Based on Taylor Series
11.2.3 Midpoint Method
11.2.4 Methods Based on Quadrature Formulas
11.2.5 Classical Fourth-Order Runge–Kutta and Runge–Kutta–Fehlberg Methods
11.2.6 An Example Using MATLAB’s ODE Solver
11.2.7 Analysis of One-Step Methods
11.2.8 Practical Implementation Considerations
11.2.9 Systems of Equations
11.3 Multistep Methods
11.3.1 Adams–Bashforth and Adams–Moulton Methods
11.3.2 General Linear m-Step Methods
11.3.3 Linear Difference Equations
11.3.4 The Dahlquist Equivalence Theorem
11.4 Stiff Equations
11.4.1 Absolute Stability
11.4.2 Backward Differentiation Formulas (BDF Methods)
11.4.3 Implicit Runge–Kutta (IRK) Methods
11.5 Solving Systems of Nonlinear Equations in Implicit Methods
11.5.1 Fixed Point Iteration
11.5.2 Newton’s Method
11.6 Chapter 11 Exercises
12 More Numerical Linear Algebra: Eigenvalues and Iterative Methods for Solving Linear Systems
12.1 Eigenvalue Problems
12.1.1 The Power Method for Computing the Largest Eigenpair
12.1.2 Inverse Iteration
12.1.3 Rayleigh Quotient Iteration
12.1.4 The QR Algorithm
12.1.5 Google’s PageRank
12.2 Iterative Methods for Solving Linear Systems
12.2.1 Basic Iterative Methods for Solving Linear Systems
12.2.2 Simple Iteration
12.2.3 Analysis of Convergence
12.2.4 The Conjugate Gradient Algorithm
12.2.5 Methods for Nonsymmetric Linear Systems
12.3 Chapter 12 Exercises
13 Numerical Solution of Two-Point Boundary Value Problems
13.1 An Application: Steady-State Temperature Distribution
13.2 Finite Difference Methods
13.2.1 Accuracy
13.2.2 More General Equations and Boundary Conditions
13.3 Finite Element Methods
13.3.1 Accuracy
13.4 Spectral Methods
13.5 Chapter 13 Exercises
14 Numerical Solution of Partial Differential Equations
14.1 Elliptic Equations
14.1.1 Finite Difference Methods
14.1.2 Finite Element Methods
14.2 Parabolic Equations
14.2.1 Semidiscretization and the Method of Lines
14.2.2 Discretization in Time
14.3 Separation of Variables
14.3.1 Separation of Variables for Difference Equations
14.4 Hyperbolic Equations
14.4.1 Characteristics
14.4.2 Systems of Hyperbolic Equations
14.4.3 Boundary Conditions
14.4.4 Finite Difference Methods
14.5 Fast Methods for Poisson’s Equation
14.5.1 The Fast Fourier Transform
14.6 Multigrid Methods
14.7 Chapter 14 Exercises
Appendix A Review of Linear Algebra
A.1 Vectors and Vector Spaces
A.2 Linear Independence and Dependence
A.3 Span of a Set of Vectors; Bases and Coordinates; Dimension of a Vector Space
A.4 The Dot Product; Orthogonal and Orthonormal Sets; the Gram–Schmidt Algorithm
A.5 Matrices and Linear Equations
A.6 Existence and Uniqueness of Solutions; the Inverse; Conditions for Invertibility
A.7 Linear Transformations; the Matrix of a Linear Transformation
A.8 Similarity Transformations; Eigenvalues and Eigenvectors
Appendix B Taylor’s Theorem in Multidimensions
References
Index
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