Numerical Methods Using Matlab

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  • Edition: 3rd
  • Format: Paperback
  • Copyright: 2012-07-13
  • Publisher: Academic Pr

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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

Prefacep. xiii
List of Figuresp. xv
An Introduction to MatlabŪp. 1
The Matlab Software Packagep. 1
Matrices and Matrix Operations in Matlabp. 3
Manipulating the Elements of a Matrixp. 5
Transposing Matricesp. 8
Special Matricesp. 9
Generating Matrices and Vectors with Specified Element Valuesp. 10
Matrix Functionsp. 13
Using the Matlab \ Operator for Matrix Divisionp. 14
Element-by-Element Operationsp. 14
Scalar Operations and Functionsp. 15
String Variablesp. 19
Input and Output in Matlabp. 24
Matlab Graphicsp. 27
Three-Dimensional Graphicsp. 34
Manipulating Graphics-Handle Graphicsp. 35
Scripting in Matlabp. 43
User-Defined Functions in Matlabp. 49
Data Structures in Matlabp. 53
Editing Matlab Scriptsp. 57
Some Pitfalls in Matlabp. 59
Faster Calculations in Matlabp. 60
Problemsp. 61
Linear Equations and Eigensystemsp. 67
Introductionp. 67
linear Equation Systemsp. 70
Operators \ and / for Solving Ax = bp. 75
Accuracy of Solutions and Ill-Conditioningp. 80
Elementary Row Operationsp. 83
Solution of Ax = b by Gaussian Eliminationp. 84
LU Decompositionp. 86
Cholesky Decompositionp. 91
QR Decompositionp. 93
Singular Value Decompositionp. 97
The Pseudo-Inversep. 100
Over- and Underdetermined Systemsp. 106
Iterative Methodsp. 114
Sparse Matricesp. 115
The Eigenvalue Problemp. 126
Iterative Methods for Solving the Eigenvalue Problemp. 130
The Matlab Function eigp. 135
Summaryp. 139
Problemsp. 140
Solution of Nonlinear Equationsp. 147
Introductionp. 147
The Nature of Solutions to Nonlinear Equationsp. 149
The Bisection Algorithmp. 150
Iterative or Fixed Point Methodsp. 151
The Convergence of Iterative Methodsp. 152
Ranges for Convergence and Chaotic Behaviorp. 153
Newton's Methodp. 156
Schroder's Methodp. 160
Numerical Problemsp. 162
The Matlab Function fzero and Comparative Studiesp. 164
Methods for Finding All the Roots of a Polynomialp. 166
Solving Systems of Nonlinear Equationsp. 171
Broyden's Method for Solving Nonlinear Equationsp. 175
Comparing the Newton and Broyden Methodsp. 178
Summaryp. 178
Problemsp. 179
Differentiation and Integrationp. 185
Introductionp. 185
Numerical Differentiationp. 185
Numerical Integrationp. 189
Simpson's Rulep. 190
Newton-Cotes Formulaep. 194
Romberg Integrationp. 196
Gaussian Integrationp. 198
Infinite Ranges of Integrationp. 201
Gauss-Chebyshev Formulap. 206
Gauss-Lobatto Integrationp. 207
Filon's Sine and Cosine Formulaep. 211
Problems in the Evaluation of Integralsp. 215
Test Integralsp. 217
Repeated Integralsp. 219
Matlab Functions for Double and Triple Integrationp. 224
Summaryp. 225
Problemsp. 226
Solution of Differential Equationsp. 233
Introductionp. 233
Euler's Methodp. 235
The Problem of Stabilityp. 237
The Trapezoidal Methodp. 239
Runge-Kutta Methodsp. 242
Predictor-Corrector Methodsp. 246
Hamming's Method and the Use of Error Estimatesp. 249
Error Propagation in Differential Equationsp. 251
The Stability of Particular Numerical Methodsp. 252
Systems of Simultaneous Differential Equationsp. 256
The Lorenz Equationsp. 259
The Predator-Prey Problemp. 260
Differential Equations Applied to Neural Networksp. 262
Higher-Order Differential Equationsp. 266
Stiff Equationsp. 267
Special Techniquesp. 270
Extrapolation Techniquesp. 274
Summaryp. 276
Problemsp. 276
Boundary Value Problemsp. 283
Classification of Second-Order Partial Differential Equationsp. 283
The Shooting Methodp. 284
The Finite Difference Methodp. 287
Two-Point Boundary Value Problemsp. 289
Parabolic Partial Differential Equationsp. 295
Hyperbolic Partial Differential Equationsp. 299
Elliptic Partial Differential Equationsp. 302
Summaryp. 309
Problemsp. 310
Fitting Functions to Datap. 313
Introductionp. 313
Interpolation Using Polynomialsp. 313
Interpolation Using Splinesp. 317
Fourier Analysis of Discrete Datap. 321
Multiple Regression: Least Squares Criterionp. 335
Diagnostics for Model Improvementp. 339
Analysis of Residualsp. 343
Polynomial Regressionp. 347
Fitting General Functions to Datap. 355
Nonlinear Least Squares Regressionp. 356
Transforming Datap. 359
Summaryp. 363
Problemsp. 363
Optimization Methodsp. 371
Introductionp. 371
Linear Programming Problemsp. 371
Optimizing Single-Variable Functionsp. 378
The Conjugate Gradient Methodp. 382
Moller's Scaled Conjugate Gradient Methodp. 388
Conjugate Gradient Method for Solving Linear Systemsp. 394
Genetic Algorithmsp. 397
Continuous Genetic Algorithmp. 413
Simulated Annealingp. 418
Constrained Nonlinear Optimizationp. 421
The Sequential Unconstrained Minimization Techniquep. 426
Summaryp. 429
Problemsp. 429
Applications of the Symbolic Toolboxp. 433
Introduction to the Symbolic Toolboxp. 433
Symbolic Variables and Expressionsp. 434
Variable-Precision Arithmetic in Symbolic Calculationsp. 439
Series Expansion and Summationp. 441
Manipulation of Symbolic Matricesp. 444
Symbolic Methods for the Solution of Equationsp. 449
Special Functionsp. 450
Symbolic Differentiationp. 452
Symbolic Partial Differentiationp. 454
Symbolic Integrationp. 456
Symbolic Solution of Ordinary Differential Equationsp. 459
The Laplace Transformp. 464
The Z-Transformp. 466
Fourier Transform Methodsp. 468
Linking Symbolic and Numerical Processesp. 472
Summaryp. 475
Problemsp. 475
Matrix Algebrap. 481
Introductionp. 481
Matrices and Vectorsp. 481
Some Special Matricesp. 482
Determinantsp. 483
Matrix Operationsp. 484
Complex Matricesp. 485
Matrix Propertiesp. 486
Some Matrix Relationshipsp. 486
Eigenvaluesp. 487
Definition of Normsp. 487
Reduced Row Echelon Formp. 488
Differentiating Matricesp. 489
Square Root of a Matrixp. 490
Error Analysisp. 491
Introductionp. 491
Errors in Arithmetic Operationsp. 492
Errors in the Solution of Linear Equation Systemsp. 493
Solutions to Selected Problemsp. 497
Bibliographyp. 521
Indexp. 525
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