Numerical Methods: Design, Analysis and Computer Implementation of Algorithms

Machine generated contents note: 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 Contents note continued: 3.3.1.Buffon's Needle 3.3.2.Estimating ? 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 Contents note continued: 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 Contents note continued: 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 Contents note continued: 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 Contents note continued: 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 Contents note continued: 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 Contents note continued: 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 Multidimnsions.