Numerical Methods for Engineers and Scientists_ An Introduction with Applications using MATLAB

Cover......Page 1
Preface......Page 5
Contents......Page 13
1.1 Background......Page 19
1.2 Representation of Numbers on a Computer......Page 22
1.3 Errors in Numerical Solutions......Page 28
1.4 Computers and Programming......Page 33
1.5 Problems......Page 36
2.1 Background......Page 41
2.2 Concepts from Pre-Calculus and Calculus......Page 42
2.3 Vectors......Page 46
2.4 Matrices and Linear Algebra......Page 50
2.5 Ordinary Differential Equations (ODE)......Page 59
2.6 Functions of Two or More Independent Variable......Page 62
2.7 Taylor Series Expansion of Functions......Page 65
2.8 Inner Product and Orthogonality......Page 68
2.9 Problems......Page 69
3.1 Background......Page 75
3.2 Estimation of Errors in Numerical Solutions......Page 77
3.3 Bisection Method......Page 79
3.4 Regula Falsi Method......Page 82
3.5 Newton's Method......Page 84
3.6 Secant Method......Page 89
3.7 Fixed-Point Iteration Method......Page 92
3.8 Use of MATLAB Built-In Functions for Solving Nonlinear Equations......Page 95
3.9 Equations with Multiple Solutions......Page 97
3.10 Systems of Nonlinear Equations......Page 99
3.11 Problems......Page 106
4.1 Background......Page 117
4.2 Gauss Elimination Method......Page 120
4.3 Gauss Elimination with Pivoting......Page 130
4.4 Gauss-Jordan Elimination Method......Page 133
4.5 LU Decomposition Method......Page 136
4.6 Inverse of a Matrix......Page 146
4.7 Iterative Methods......Page 150
4.8 Use of MATLAB Built-In Functions for Solving a System of Linear Equations......Page 154
4.9 Tridiagonal Systems of Equations......Page 159
4.10 Error, Residual, Norms, and Condition Number......Page 164
4.11 ILL-Conditioned Systems......Page 171
4.12 Problems......Page 173
5.1 Background......Page 183
5.3 The Basic Power Method......Page 185
5.4 The Inverse Power Method......Page 190
5.5 The Shifted Power Method......Page 191
5.6 The QR Factorization and Iteration Method......Page 192
5.7 Use of MATLAB Built-In Functions for Determining Eigenvalues and Eigenvectors......Page 202
5.8 Problems......Page 204
6.1 Background......Page 211
6.2 Curve Fitting with a Linear Equation......Page 213
6.3 Curve Fitting with Nonlinear Equation by Writing the Equation in a Linear Form......Page 219
6.4 Curve Fitting with Quadratic and Higher-Order Polynomials......Page 223
6.5 Interpolation Using a Single Polynomial......Page 228
6.6 Piecewise (Spline) Interpolation......Page 241
6.7 Use of MATLAB Built-In Functions for Curve Fitting and Interpolation......Page 254
6.8 Curve Fitting with a Linear Combination of Nonlinear Functions......Page 256
6.9 Problems......Page 259
7.1 Background......Page 269
7.2 Approximating a Square Wave by a Series of Sine Functions......Page 271
7.3 General (Infinite) Fourier Series......Page 275
7.4 Complex Form of the Fourier Series......Page 279
7.5 The Discrete Fourier Series and Discrete Fourier Transform......Page 281
7.6 Complex Discrete Fourier Transform......Page 286
7.7 Power (Energy) Spectrum......Page 289
7.8 Aliasing and Nyquist Frequency......Page 290
7.10 Use of MATLAB Built-In Functions for Calculating Discrete Fourier Transform......Page 296
7 .11 Leakage and Windowing......Page 302
7.12 Bandwidth and Filters......Page 304
7.13 The Fast Fourier Transform (FF T)......Page 307
7.14 Problems......Page 316
8.1 Background......Page 321
8.2 Finite Difference Approximation of the Derivative......Page 323
8.3 Finite Difference Formulas Using Taylor Series Expansion......Page 328
8.4 Summary of Finite Difference Formulas for Numerical Differentiation......Page 335
8.5 Differentiation Formulas Using Lagrange Polynomials......Page 337
8.7 Use of MATLAB Built-In Functions for Numerical Differentiation......Page 338
8.8 Richardson's Extrapolation......Page 340
8.9 Error in Numerical Differentiation......Page 343
8.10 Numerical Partial Differentiation......Page 345
8.11 Problems......Page 348
9.1 Background......Page 359
9.2 Rectangle and Midpoint Methods......Page 362
9.3 Trapezoidal Method......Page 364
9.4 Simpson's Methods......Page 368
9.5 Gauss Quadrature......Page 373
9.6 Evaluation of Multiple Integrals......Page 378
9.7 Use of MATLAB Built-In Functions for Integration......Page 380
9.8 Estimation of Error in Numerical Integration......Page 382
9.9 Richardson's Extrapolation......Page 384
9 .10 Romberg Integration......Page 387
9.11 Improper Integrals......Page 390
9.12 Problems......Page 392
10.1 Background......Page 403
10.2 Euler's Methods......Page 408
10.3 Modified Euler's Method......Page 419
10.4 Midpoint Method......Page 422
10.5 Runge-Kutta Methods......Page 423
10.6 Multistep Methods......Page 435
10. 7 Predictor-Corrector Methods......Page 438
10.8 System of First-Order Ordinary Differential Equations......Page 440
10.9 Solving a Higher-Order Initial Value Problem......Page 450
10.10 Use of MATLAB Built-In Functions for Solving Initial-Value Problems......Page 455
10.11 Local Truncation Error in Second-Order Range-Kutta Method......Page 465
10.12 Step Size for Desired Accuracy......Page 466
10.13 Stability......Page 470
10.14 Stiff Ordinary Differential Equations......Page 472
10.15 Problems......Page 475
11.1Background......Page 489
11.2 The Shooting Method......Page 492
11.3 Finite Difference Method......Page 500
11.4 Use of MATLAB Built-In Functions for Solving Boundary Value Problems......Page 510
11.5 Error and Stability in Numerical Solution of Boundary Value Problems......Page 515
11.6 Problems......Page 517
A.2 Starting with MATLAB......Page 527
A.3 Arrays......Page 532
A.4 Mathematical Operations with Arrays......Page 537
A.5 Script Files......Page 542
A.6 Plotting......Page 544
A.7 User-Defined Functions and Function Files......Page 546
A.8 Anonymous Functions......Page 548
A.9 Function functions......Page 550
A.10 Subfunctions......Page 553
A.11 Programming in MATLAB......Page 555
A.12 Problems......Page 560
Appendix B. MATLAB Programs......Page 565
C.1 Orthogonality of Sines and Cosines for Discrete Points......Page 569
C.2 Determination of the Real DF T......Page 571
Index......Page 573