Numerical Methods

by Lindfield; Penny

• ISBN13: 9780123869425
• ISBN10: 0123869420
• Edition: 3rd
• Format: Paperback
• Copyright: 2012-07-13
• Publisher: Elsevier Science

Summary

Numerical Methods using MATLAB, 3rd edition is an extensive reference offering hundreds of useful and important numerical algorithms that can be implemented into MATLAB for a graphical interpretation to help researchers analyze a particular outcome. Many worked examples are given together with exercises and solutions to illustrate how numerical methods can be used to study problems which have applications in the biosciences, chaos, optimization, engineering and science across the board. Over 500 numerical algorithms, their fundamental principles, and applications Graphs are used extensively to clarify the complexity of problems Includes coded genetic algorithms Includes the lagrange multiplier method User-friendly and written in a conversational style

Author Biography

George Lindfield is a former lecturer in Mathematics and Computing at the School of Engineering and Applied Science, Aston University in the United Kingdom. John Penny is an Emeritus Professor of Mechanical Engineering at the School of Engineering and Applied Science, Aston University in the United Kingdom.

Table of Contents

Preface	p. xiii
List of Figures	p. xv
An Introduction to Matlab®	p. 1
The Matlab Software Package	p. 1
Matrices and Matrix Operations in Matlab	p. 3
Manipulating the Elements of a Matrix	p. 5
Transposing Matrices	p. 8
Special Matrices	p. 9
Generating Matrices and Vectors with Specified Element Values	p. 10
Matrix Functions	p. 13
Using the Matlab \ Operator for Matrix Division	p. 14
Element-by-Element Operations	p. 14
Scalar Operations and Functions	p. 15
String Variables	p. 19
Input and Output in Matlab	p. 24
Matlab Graphics	p. 27
Three-Dimensional Graphics	p. 34
Manipulating Graphics-Handle Graphics	p. 35
Scripting in Matlab	p. 43
User-Defined Functions in Matlab	p. 49
Data Structures in Matlab	p. 53
Editing Matlab Scripts	p. 57
Some Pitfalls in Matlab	p. 59
Faster Calculations in Matlab	p. 60
Problems	p. 61
Linear Equations and Eigensystems	p. 67
Introduction	p. 67
linear Equation Systems	p. 70
Operators \ and / for Solving Ax = b	p. 75
Accuracy of Solutions and Ill-Conditioning	p. 80
Elementary Row Operations	p. 83
Solution of Ax = b by Gaussian Elimination	p. 84
LU Decomposition	p. 86
Cholesky Decomposition	p. 91
QR Decomposition	p. 93
Singular Value Decomposition	p. 97
The Pseudo-Inverse	p. 100
Over- and Underdetermined Systems	p. 106
Iterative Methods	p. 114
Sparse Matrices	p. 115
The Eigenvalue Problem	p. 126
Iterative Methods for Solving the Eigenvalue Problem	p. 130
The Matlab Function eig	p. 135
Summary	p. 139
Problems	p. 140
Solution of Nonlinear Equations	p. 147
Introduction	p. 147
The Nature of Solutions to Nonlinear Equations	p. 149
The Bisection Algorithm	p. 150
Iterative or Fixed Point Methods	p. 151
The Convergence of Iterative Methods	p. 152
Ranges for Convergence and Chaotic Behavior	p. 153
Newton's Method	p. 156
Schroder's Method	p. 160
Numerical Problems	p. 162
The Matlab Function fzero and Comparative Studies	p. 164
Methods for Finding All the Roots of a Polynomial	p. 166
Solving Systems of Nonlinear Equations	p. 171
Broyden's Method for Solving Nonlinear Equations	p. 175
Comparing the Newton and Broyden Methods	p. 178
Summary	p. 178
Problems	p. 179
Differentiation and Integration	p. 185
Introduction	p. 185
Numerical Differentiation	p. 185
Numerical Integration	p. 189
Simpson's Rule	p. 190
Newton-Cotes Formulae	p. 194
Romberg Integration	p. 196
Gaussian Integration	p. 198
Infinite Ranges of Integration	p. 201
Gauss-Chebyshev Formula	p. 206
Gauss-Lobatto Integration	p. 207
Filon's Sine and Cosine Formulae	p. 211
Problems in the Evaluation of Integrals	p. 215
Test Integrals	p. 217
Repeated Integrals	p. 219
Matlab Functions for Double and Triple Integration	p. 224
Summary	p. 225
Problems	p. 226
Solution of Differential Equations	p. 233
Introduction	p. 233
Euler's Method	p. 235
The Problem of Stability	p. 237
The Trapezoidal Method	p. 239
Runge-Kutta Methods	p. 242
Predictor-Corrector Methods	p. 246
Hamming's Method and the Use of Error Estimates	p. 249
Error Propagation in Differential Equations	p. 251
The Stability of Particular Numerical Methods	p. 252
Systems of Simultaneous Differential Equations	p. 256
The Lorenz Equations	p. 259
The Predator-Prey Problem	p. 260
Differential Equations Applied to Neural Networks	p. 262
Higher-Order Differential Equations	p. 266
Stiff Equations	p. 267
Special Techniques	p. 270
Extrapolation Techniques	p. 274
Summary	p. 276
Problems	p. 276
Boundary Value Problems	p. 283
Classification of Second-Order Partial Differential Equations	p. 283
The Shooting Method	p. 284
The Finite Difference Method	p. 287
Two-Point Boundary Value Problems	p. 289
Parabolic Partial Differential Equations	p. 295
Hyperbolic Partial Differential Equations	p. 299
Elliptic Partial Differential Equations	p. 302
Summary	p. 309
Problems	p. 310
Fitting Functions to Data	p. 313
Introduction	p. 313
Interpolation Using Polynomials	p. 313
Interpolation Using Splines	p. 317
Fourier Analysis of Discrete Data	p. 321
Multiple Regression: Least Squares Criterion	p. 335
Diagnostics for Model Improvement	p. 339
Analysis of Residuals	p. 343
Polynomial Regression	p. 347
Fitting General Functions to Data	p. 355
Nonlinear Least Squares Regression	p. 356
Transforming Data	p. 359
Summary	p. 363
Problems	p. 363
Optimization Methods	p. 371
Introduction	p. 371
Linear Programming Problems	p. 371
Optimizing Single-Variable Functions	p. 378
The Conjugate Gradient Method	p. 382
Moller's Scaled Conjugate Gradient Method	p. 388
Conjugate Gradient Method for Solving Linear Systems	p. 394
Genetic Algorithms	p. 397
Continuous Genetic Algorithm	p. 413
Simulated Annealing	p. 418
Constrained Nonlinear Optimization	p. 421
The Sequential Unconstrained Minimization Technique	p. 426
Summary	p. 429
Problems	p. 429
Applications of the Symbolic Toolbox	p. 433
Introduction to the Symbolic Toolbox	p. 433
Symbolic Variables and Expressions	p. 434
Variable-Precision Arithmetic in Symbolic Calculations	p. 439
Series Expansion and Summation	p. 441
Manipulation of Symbolic Matrices	p. 444
Symbolic Methods for the Solution of Equations	p. 449
Special Functions	p. 450
Symbolic Differentiation	p. 452
Symbolic Partial Differentiation	p. 454
Symbolic Integration	p. 456
Symbolic Solution of Ordinary Differential Equations	p. 459
The Laplace Transform	p. 464
The Z-Transform	p. 466
Fourier Transform Methods	p. 468
Linking Symbolic and Numerical Processes	p. 472
Summary	p. 475
Problems	p. 475
Appendices
Matrix Algebra	p. 481
Introduction	p. 481
Matrices and Vectors	p. 481
Some Special Matrices	p. 482
Determinants	p. 483
Matrix Operations	p. 484
Complex Matrices	p. 485
Matrix Properties	p. 486
Some Matrix Relationships	p. 486
Eigenvalues	p. 487
Definition of Norms	p. 487
Reduced Row Echelon Form	p. 488
Differentiating Matrices	p. 489
Square Root of a Matrix	p. 490
Error Analysis	p. 491
Introduction	p. 491
Errors in Arithmetic Operations	p. 492
Errors in the Solution of Linear Equation Systems	p. 493
Solutions to Selected Problems	p. 497
Bibliography	p. 521
Index	p. 525
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