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Numerical Methods for Engineers and Scientists, 3rd Edition provides engineers with a more concise treatment of the essential topics of numerical methods while emphasizing MATLAB use.
List of contents
Preface iii
Chapter 1 Introduction 1
1.1 Background 1
1.2 Representation of Numbers on a Computer 4
1.3 Errors in Numerical Solutions 10
1.4 Computers and Programming 15
1.5 Problems 18
Chapter 2 Mathematical Background 23
2.1 Background 23
2.2 Concepts from Pre-Calculus and Calculus 24
2.3 Vectors 28
2.4 Matrices and Linear Algebra 32
2.5 Ordinary Differential Equations (ODE) 41
2.6 Functions of Two or More Independent Variables 44
2.7 Taylor Series Expansion of Functions 47
2.8 Inner Product and Orthogonality 50
2.9 Problems 51
Chapter 3 Solving Nonlinear Equations 57
3.1 Background 57
3.2 Estimation of Errors in Numerical Solutions 59
3.3 Bisection Method 61
3.4 Regula Falsi Method 64
3.5 Newton's Method 66
3.6 Secant Method 71
3.7 Fixed-Point Iteration Method 74
3.8 Use of MATLAB Built-In Functions for Solving Nonlinear Equations 77
3.9 Equations with Multiple Solutions 79
3.10 Systems of Nonlinear Equations 81
3.11 Problems 88
Chapter 4 Solving a System of Linear Equations 99
4.1 Background 99
4.2 Gauss Elimination Method 102
4.3 Gauss Elimination with Pivoting 112
4.4 Gauss-Jordan Elimination Method 115
4.5 LU Decomposition Method 118
4.6 Inverse of a Matrix 128
4.7 Iterative Methods 132
4.8 Use of MATLAB Built-In Functions for Solving a System of Linear Equations 136
4.9 Tridiagonal Systems of Equations 141
4.10 Error, Residual, Norms, and Condition Number 146
4.11 Ill-Conditioned Systems 151
4.12 Problems 155
Chapter 5 Eigenvalues and Eigenvectors 165
5.1 Background 165
5.2 The Characteristic Equation 167
5.3 The Basic Power Method 167
5.4 The Inverse Power Method 172
5.5 The Shifted Power Method 173
5.6 The QR Factorization and Iteration Method 174
5.7 Use of MATLAB Built-In Functions for Determining Eigenvalues and
Eigenvectors 184
5.8 Problems 186
Chapter 6 Curve Fitting and Interpolation 193
6.1 Background 193
6.2 Curve Fitting with a Linear Equation 195
6.3 Curve Fitting with Nonlinear Equation by Writing the Equation in a Linear Form 201
6.4 Curve Fitting with Quadratic and Higher-Order Polynomials 205
6.5 Interpolation Using a Single Polynomial 210
6.6 Piecewise (Spline) Interpolation 223
6.7 Use of MATLAB Built-In Functions for Curve Fitting and Interpolation 236
6.8 Curve Fitting with a Linear Combination of Nonlinear Functions 238
6.9 Problems 241
Chapter 7 Fourier Methods 251
7.1 Background 251
7.2 Approximating a Square Wave by a Series of sine functions 254
7.3 General (Infinite) Fourier Series 257
7.4 Complex Form of the Fourier Series 262
7.5 The Discrete Fourier Series and Discrete Fourier transform 263
7.6 Complex Discrete Fourier Transform 269
7.7 Power (Energy) Spectrum 272
7.8 Aliasing and Nyquist Frequency 273
7.9 Alternative Forms of the Discrete Fourier Transform 278
7.10 Use of MATLAB Built-In Functions for Calculating Discrete Fourier Transform 279
7.11 Leakage and Windowing 284
7.12 Bandwidth and Filters 286
7.13 The Fast Fourier Transform (FFT) 288
7.14 Problems 298
Chapter 8 Numerical Differentiation 303
8.1 Background 303
8.2 Finite Difference Approximation of the Derivative 305
8.3 Finite Difference Formulas Using Taylor Series Expansion 310
8.4 Summary of Finite Difference Formulas for Numerical Differentiation 317
8.5 Differentiation Formulas Using Lagrange Polynomials 319
8.6 Differentiation Using Curve Fitting 320
8.7 Use of MATLAB Built-In Functions for Numerical Differentiation 320
8.8 Richardson's Extrapolation 322
8.9 Error in Numerical Differentiation 325
8.10 Numerical Partial Differentiation 327
8.11 Problems 330
Chapter 9 Numerical Integration 341
9.1 Background 341
9.1.1 Overview of Approaches in Numerical Integration 342
9.2 Rectangle and Midpoint Methods 344
9.3 Trapezoidal Method 346
9.4 Simpson's
Summary
Numerical Methods for Engineers and Scientists, 3rd Edition provides engineers with a more concise treatment of the essential topics of numerical methods while emphasizing MATLAB use.
