Numerical Methods for Engineers. and Scientists. Applications using MATLAB. An Introduction with. Vish- Subramaniam. Third Edition. Amos Gilat.

Transcription

1 Numerical Methods for Engineers An Introduction with and Scientists Applications using MATLAB Third Edition Amos Gilat Vish- Subramaniam Department of Mechanical Engineering The Ohio State University Wiley

2 Contents Preface iii Chapter 1 Introduction l 1.1 Background Representation of Numbers on a Computer Errors in Numerical Solutions Round-OffErrors Truncation Errors Total Error Computers and Programming Problems 18 Chapter 2 Mathematical Background Background Concepts from Pre-Calculus and Calculus Vectors Operations with Vectors Matrices and Linear Algebra Operations SpecialMatrices '35 with Matrices Inverse ofa Matrix Properties ofmatrices Determinant of a Matrix Cramer's Rule and Solution ofa System of Simultaneous Linear Equations Norms Ordinary Differential Equations (ODE) Functions of Two or More Independent Definition ofthe Partial Derivative Chain Rule TheJacobian 46 Variables Taylor Series Expansion of Functions Taylor Seriesfor a Function ofone Variable Taylor Seriesfor a Function oftwo Variables Inner Product and Orthogonality Problems 51 xi

3 Contents Chapter 3 Solving Nonlinear Equations Background Estimation of Errors in Numerical Solutions Bisection Method Regula Falsi Method Newton's Method Secant Method Fixed-Point Iteration Method Use ofmatlab Built-in Functions for Solving Nonlinear Equations The f zero Command The roots Command Equations with Multiple Solutions Systems ofnonlinear Equations Newton's Methodfor Solving a System of Nonlinear Equations Fixed-Point Iteration Methodfor Solving a System ofnonlinear Equations Problems 88 Chapter 4 Solving a System oflinear Equations Background Overview ofnumerical Methods for Solving a System oflinear Algebraic Equations Gauss Elimination Method Potential Difficulties When Applying the Gauss Elimination Method Gauss Elimination with Pivoting Gauss-Jordan Elimination Method LU Decomposition Method LU Decomposition Using the Gauss Elimination Procedure LU Decomposition Using Crout's Method LU Decomposition with Pivoting Inverse ofa Matrix Calculating the Inverse with the LU Decomposition Method Calculating the Inverse Using the Gauss-Jordan Method Iterative Methods JacobiIterative Method Gauss-Seidel Iterative Method Use ofmatlab Built-in Functions for Solving a System oflinear Equations 136 Division Solving a System ofequations Using MATLAB's Left and Right Solving a System of Equations Using MATLAB"s Inverse Operation MATLAB's Built-in Function forludecomposition Additional MATLAB Built-in Functions Tridiagonal Systems of Equations 141

4 Contents xiii 4.10 Error, Residual, Norms, and Condition Number Error and Residual Norms and Condition Number Ill-Conditioned Systems Problems 155 Chapter 5 Eigenvalues and Eigenvectors Background The Characteristic Equation The Basic Power Method The Inverse Power Method The Shifted Power Method The QR Factorization and Iteration Method Use ofmatlab Built-in Functions for Determining Eigenvalues and Eigenvectors Problems 186 Chapter 6 Curve Fitting and Interpolation Background Curve Fitting with a Linear Equation Measuring How Good Is a Fit Linear Least-Squares Regression Curve Fitting with Nonlinear Equation by Writing the Equation in a Linear Form Curve Fitting with Quadratic and Higher-Order Polynomials Interpolation Using a Single Polynomial Lagrange Interpolating Polynomials Newton's Interpolating Polynomials Piecewise (Spline) Interpolation Linear Splines Quadratic Splines Cubic Splines Use of MATLAB Built-in Functions for Curve Fitting and Interpolation Curve Fitting with a Linear Combination ofnonlinear Functions Problems 241 Chapter 7 Fourier Methods Background Approximating a Square Wave by a Series of Sine Functions General (Infinite) Fourier Series Complex Form of the Fourier Series 261

5 xiv Contents 7.5 The Discrete Fourier Series and Discrete Fourier Transform Complex Discrete Fourier Transform Power (Energy) Spectrum Aliasing and Nyquist Frequency Alternative Forms of the Discrete Fourier Transform Use of MATLAB Built-in Functions for Calculating Discrete Fourier Transform Leakage and Windowing Bandwidth and Filters The Fast Fourier Transform (FFT) Problems 298 Chapter 8 Numerical Differentiation Background Finite Difference Approximation ofthe Derivative Finite Difference Formulas Using Taylor Series Expansion Finite Difference Formulas offirst Derivative Finite Difference Formulasfor the Second Derivative Summary of Finite Difference Formulas for Numerical Differentiation Differentiation Formulas Using Lagrange Polynomials Differentiation Using Curve Fitting Use of MATLAB Built-in Functions for Numerical Differentiation Richardson's Extrapolation Error in Numerical Differentiation Numerical Partial Differentiation Problems 330 Chapter 9 Numerical Integration Background Overview ofapproaches in Numerical Integration Rectangle and Midpoint Methods Trapezoidal Method Composite Trapezoidal Method Simpson's Methods Simpson's 1/3 Method Simpson's3/8Method Gauss Quadrature Evaluation ofmultiple Integrals Use ofmatlab Built-in Functions for Integration Estimation oferror in Numerical Integration Richardson's Extrapolation 366

6 , Contents xv 9.10 Romberg Integration Improper Integrals Integrals with Singularities 372 Integrals with Unbounded Limits Problems 374 Chapter 10 Ordinary Differential Equations: Initial-Value Problems Background Euler's Methods Euler 's Explicit Method Analysis oftruncation Error in Euler's Explicit Method Euler's Implicit Method Modified Euler's Method Midpoint 10.5 Runge-Kutta Method 404 Methods Second-Order Runge-Kutta Methods Third-Order Runge-Kutta Methods Fourth-Order Runge-Kutta Methods Multistep Methods Adams-BashforthMethod Adams-Moulton Method Predictor-Corrector Methods System of First-Order Ordinary Differential Equations Solving a System of First-Order ODEs Using Euler's Explicit Method Solving a System of First-Order ODEs Using Second-Order Runge-Kutta Method (Modified Euler Version) Solving a System offirst-order ODEs Using the Classical Fourth-Order Runge-Kutta Method Solving a Higher-Order Initial Value Problem Use of MATLAB Built-in Functions for Solving Solving a Single First-Order ODE Using MATLAB Solving a System of First-Order ODEs Using MATLAB 444 Initial-Value Problems Local Truncation Error in Second-Order Range-Kutta Method Step Size for Desired Accuracy Stability Stiff Ordinary Differential Equations Problems 457

7 xvi Contents Chapter 11 Ordinary Differential Equations: Boundary- Value Problems Background The Shooting Method Finite Difference Method Use of MATLAB Built-in Functions for Solving Boundary Value Problems Error and Stability in Numerical Solution of Boundary 11.6 Problems 499 Value Problems 497 Appendix A Introductory MATLAB 509 A. 1 Background 509 A.2 Starting with MATLAB 509 A.3 Arrays 514 A.4 Mathematical Operations with Arrays 519 A.5 Script Files 524 A.6 Plotting 526 A.7 User-Defined Functions and Function Files 528 A. 8 Anonymous Functions 530 A.9 Function functions 532 A. 10 Subfunctions 535 A. 11 Programming in MATLAB 537 A Relational andlogical Operators 537 A.11.2 Conditional Statements, if-else Structures 538 A Loops 541 A. 12 Problems 542 Appendix B MATLAB Programs 547 Appendix C Derivation of the Real Discrete Fourier Transform (DFT) 551 C. 1 Orthogonality of Sines and Cosines for Discrete Points 551 C.2 Determination ofthe Real DFT 553 Index 555