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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