Applied Numerical Analysis

Transcription

1 Applied Numerical Analysis Using MATLAB Second Edition Laurene V. Fausett Texas A&M University-Commerce PEARSON Prentice Hall Upper Saddle River, NJ 07458

2 Contents Preface xi 1 Foundations Introductory Examples Nonlinear Equations Linear Systems Numerical Integration Useful Background Results from Calculus Results from Linear Algebra A Little Information about Computers Some Basic Issues ' Error Convergence Getting Better Results UsingMATLAB Command Window Computations M-Files Programming in MATLAB Matrix Multiplication Chapter Wrap-Up 41 2 Functions of One Variable Bisection Method Secant-Type Methods Regula Falsi Secant Method Analysis Newton's Method Muller's Method Minimization Golden-Section Search Brent's Method Beyond the Basics Using MATLAB's Functions Laguerre's Method Zeros of a Nonlinear Function Chapter Wrap-Up 88 v 1

3 vi 3 Solving Linear Systems: Direct Methods Gaussian Elimination Basic Method Row Pivoting Gauss-Jordan Inverse of a Matrix Tridiagonal Systems Further Topics MATLAB's Methods Condition of a Matrix Iterative Refinement Chapter Wrap-Up LU and QR Factorization LU Factorization Using Gaussian Elimination Direct LU Factorization Applications Matrix Transformations Householder Transformation Givens Rotations QR Factorization Using Householder Transformations Using Givens Rotations Beyond the Basics LU Factorization with Implicit Row Pivoting EfRcient Conversion to Hessenberg Form Using MATLAB's Functions Chapter Wrap-Up Eigenvalues and Eigenvectors Power Method Basic Power Method Rayleigh Quotient Shifted Power Method Accelerating Convergence Inverse Power Method General Inverse Power Method Convergence QR Method Basic QR Method Better QR Method Finding Eigenvectors Accelerating Convergence Further Topics Singular Value Decomposition MATLAB's Methods Chapter Wrap-Up 204

4 K 6 Solving Linear Systems: Iterative Methods Jacobi Method Gauss-Seidel Method Successive Over-Relaxation Beyond the Basics MATLAB's Built-In Functions Conjugate Gradient Methods GMRES Simplex Method Chapter Wrap-Up Nonlinear Functions of Several Variables Nonlinear Systems Newton's Method Secant Methods Fixed-Point Iteration Minimization Descent Methods Quasi-Newton Methods Further Topics Levenberg-Marquardt Method Nelder-Mead Simplex Search Chapter Wrap-Up Interpolation Polynomial Interpolation Lagrange Form Newton Form Difficulties Hermite Interpolation Piecewise Polynomial Interpolation Piecewise Linear Interpolation Piecewise Quadratic Interpolation Piecewise Cubic Hermite Interpolation Cubic Spline Interpolation Beyond the Basics Rational-Function Interpolation Using MATLAB's Functions Chapter Wrap-Up Approximation Least-Squares Approximation Approximation by a Straight Line Approximation by a Parabola General Least-Squares Approximation Approximation for Other Functional Forms 348 VII

5 viii 9.2 Continuous Least-Squares Approximation Approximation Using Powers of x Orthogonal Polynomials Legendre Polynomials Chebyshev Polynomials Function Approximation at a Point Pade Approximation Taylor Approximation Further Topics Bezier Curves Using MATLAB's Functions Chapter Wrap-Up Fourier Methods Fourier Approximation and Interpolation Derivation Data on Other Intervals Radix-2 Fourier Transforms Discrete Fourier Transform Fast Fourier Transform Matrix Form of FFT Algebraic Form of FFT Mixed-Radix FFT Using MATLAB's Functions Chapter Wrap-Up Numerical Differentiation and Integration Differentiation First Derivatives Higher Derivatives Partial Derivatives Richardson Extrapolation Numerical Integration Trapezoid Rule Simpson's Rule Newton-Cotes Open Formulas Extrapolation Methods Quadrature Gaussian Quadrature Other Gauss-Type Quadratures MATLAB's Methods Differentiation Integration Chapter Wrap-Up 438

6 1 12 Ordinary Differential Equations: Fundamentals Euler's Method Geometrie Introduction Approximating the Derivative Approximating the Integral Using Taylor Series Runge-Kutta Methods Second-Order Runge-Kutta Methods Third-Order Runge-Kutta Methods Classic Runge-Kutta Method Fourth-Order Runge-Kutta Methods Fifth-Order Runge-Kutta Methods Runge-Kutta-Fehlberg Methods Multistep Methods Adams-Bashforth Methods Adams-Moulton Methods Adams Predictor-Corrector Methods Other Predictor-Corrector Methods Further Topics : MATLAB's Methods Consistency and Convergence Chapter Wrap-Up ODE: Systems, Stiffhess, Stability Systems Systems of Two ODE Euler's Method for Systems Runge-Kutta Methods for Systems Multistep Methods for Systems Second-Order ODE StifFODE BDF Methods Implicit Runge-Kutta Methods Stability A-Stable and Stiffly Stable Methods Stability in the Limit Further Topics MATLAB's Methods for Stiff ODE Extrapolation Methods Rosenbrock Methods Multivalue Methods Chapter Wrap-Up 552 IX

7 ^ 14 ODE: Boundary-Value Problems Shooting Method Linear ODE Nonlinear ODE Finite-Difference Method Linear ODE Nonlinear ODE Function Space Methods Collocation Rayleigh-Ritz Chapter Wrap-Up Partial Differential Equations Heat Equation: Parabolic PDE Explicit Method Implicit Method Crank-Nicolson Method Insulated Boundary Wave Equation: Hyperbolic PDE Explicit Method Implicit Method Poisson Equation: Elliptic PDE Finite-Element Method for Elliptic PDE Defining the Subregions Defining the Basis Functions Computing the Coefncients UsingMATLAB Chapter Wrap-Up 634 Bibliography 643 Ans wer s 653 Index 667