Numerical Methods, Algorithms and Tools in C# Book

l Introduction 1

1.1 C# and the.NET Framework 1

1.2 Installing C# and the.NET Framework 3

1.3 Overview of Object-Oriented Programming (OOP) 3

1.4 Your First C# Program 4

1.5 Overview of the IDE Debugger 9

1.6 Overview of the C# Language 11

1.6.1 Data Types 12

1.6.2 Value Types 13

1.6.3 Reference Types 14

1.6.4 Type-Parameter Types 16

1.6.5 Pointer Types 17

1.6.6 Variable Declaration 17

1.6.7 Constant Declaration 18

1.6.8 Nullable Types 18

1.6.9 Scope 18

1.6.10 Characters 18

1.6.11 Strings 19

1.6.12 Formatting of Output Data 19

1.6.13 Type Conversion 20

1.6.14 Reading Keyboard Input Data 23

1.6.15 Basic Expressions and Operators 24

1.6.16 Program Flow Mechanisms 27

1.6.17 Jump Statements 29

1.6.18 Arrays 30

1.6.19 Enumerations 32

1.6.20 Structures 32

1.6.21 Exceptions 33

1.6.22 Classes 34

Constructors and Destructors 37

Properties 38

Methods 38

1.6.23 Indexers 42

1.6.24 Overloading Methods, Constructors and Operators 42

1.6.25 Delegates 43

1.6.26 Events 46

1.6.27 Collections 57

1.6.28 File Input/Output 60

1.6.29 Output Reliability, Accuracy and Precision 65

2 The.NET Framework Math Class Library 73

2.1 Introduction 73

2.2 The.NET Framework Math Class - Fields 73

2.2.1 The Math. pi and Math. e Fields 73

2.3 The.NET Framework Math Class - Methods 74

2.3.1 The Minimum and Maximum Methods 74

2.3.2 The Power, Exponential and Logarithmic Methods 74

2.3.3 Special Multiplication, Division and Remainder Methods 76

2.3.4 The Absolute Value Method 77

2.3.5 The Sign Method 78

2.3.6 Angular Units of Measurement 78

2.3.7 The Trigonometric Functions 81

2.3.8 The Inverse Trigonometric Functions 82

2.3.9The Hyperbolic Functions 86

2.3.10 The Inverse Hyperbolic Functions 88

2.3.11 Rounding Off Numeric Data 89

The Ceiling Method 89

The Floor Method 90

The Truncation Method 90

The Round Method 91

3 Vectors and Matrices 97

3.1 Introduction 97

3.2 A Real Number Vector Library in C# 98

3.3 A Real Number Matrix Library in C# 106

4 Complex Numbers 121

4.1 Introduction 121

4.2 Fundamental Concepts 121

4.3 Complex Number Arithmetic 123

4.4 Elementary Functions of a Complex Number 125

4.4.1 Exponentials 125

4.4.2 Logarithms 125

4.4.3 Powers and Roots 127

4.4.4 Trigonometric and Hyperbolic Functions 128

4.4.5 Inverse Trigonometric and Hyperbolic Functions 130

4.5 A Complex Number Library in C# 132

4.6 A Complex Number Vector Library in C# 151

4.7 A Complex Number Matrix Library in C# 158

4.8 Generic vs. Non-Generic Coding 168

5 Sorting and Searching Algorithms 171

5.1 Introduction 171

5.2 Sorting Algorithms 172

5.3 Comparison Sorts 175

5.3.1 Bubble Sort 175

5.3.2 Cocktail Sort 178

5.3.3 Odd-Even Sort 178

5.3.4 Comb Sort 179

5.3.5 Gnome Sort 180

5.3.6 Quicksort 181

5.3.7 Insertion Sort 182

5.3.8 Shell Sort 183

5.3.9 Selection Sort 184

5.3.10 Merge Sort 185

5.3.11 Bucket Sort 186

5.3.12 Heap Sort 187

5.4 Count Sort 188

5.5 Radix Sort 189

5.6 Search Algorithms 191

5.6.1 Linear Search 192

5.6.2 Binary Search 193

5.6.3 Interpolation Search 193

5.6.4 Searching for the Maximum and Minimum Values 194

5.6.5 Searching for the N-th Largest or M-th Smallest Value 195

5.6.6 Some Useful Utilities 196

6 Bits and Bytes 199

6.1 Introduction 199

6.2 Numeric Systems 199

6.3 Bit Manipulation and Bitwise Operators 202

6.4 Assorted Bits and Bytes 223

7 Interpolation 229

7.1 Introduction 229

7.2 Linear Interpolation 230

7.3 Bilinear Interpolation 231

7.4 Polynomial Interpolation 234

7.4.1 Lagrange Interpolation 234

7.4.2 Barycentric Interpolation 236

7.4.3 Newton's Divided Differences Interpolation 238

7.5 Cubic Spline Interpolation 242

7.5.1 Natural Cubic Splines 244

7.5.2 Clamped Cubic Splines 247

8 Linear Equations 251

8.1 Introduction 251

8.2 Gaussian Elimination 253

8.3 Gauss-Jordan Elimination 254

8.4 LU Decomposition 256

8.5 Iteration Methods 259

8.5.1 Gauss-Jacobi Iteration 259

8.5.2 Gauss-Seidel Iteration 261

8.6 Eigenvalues and Jacobi's Algorithm 264

9 Nonlinear Equations 271

9.1 Introduction 271

9.2 Linear Incremental Method 272

9.3 Bisection Method 274

9.4 The Secant Method 276

9.5 False Positioning Method 277

9.6 Fixed Point Iteration 279

9.7 Newton-Raphson Method 280

10 Random Numbers 283

10.1 Introduction 283

10.2 The C# Built-In Random Number Generator 284

10.3 Other Random Number Generators 290

10.4 True Random Number Generators 295

10.5 Random Variate Generation Methods 299

10.6 Histograms 309

10.7 Random Variate Generation 312

10.7.1 Discrete Distributions 312

Bernoulli Distribution 312

Binoulli Distribution 315

Geometric Distribution 317

Negative Binomial Distribution 320

Poisson Distribution 322

Uniform Distribution (discrete) 326

10.7.2 Continuous Distributions 328

Beta Distribution 328

Beta Prime Distribution 330

Cauchy Distribution 332

Chi Distribution 334

Chi-Square Distribution 337

Erlang Distribution 340

Exponential Distribution 343

Extreme Value Distribution 345

Gamma Distribution 347

Laplace Distribution 349

Logistic Distribution 352

Lognormal Distribution 354

Normal Distribution 356

Pareto Distribution 359

Rayleigh Distribution 361

Student-t Distribution 363

Triangular Distribution 365

Uniform Distribution (continuous) 368

Weibull Distribution 370

10.8 Shuffling Algorithms 372

10.9 Adding Random Noise to Data 376

10.10 Removing Random Noise from Data 379

11 Numerical Differentiation 383

11.1 Introduction 383

11.2 Finite Difference Formulas 383

11.2.1 Forward Difference Method 385

11.2.2 Backward Difference Method 387

11.2.3 Central Difference Method 390

11.2.4 Improved Central Difference Method 392

11.3 Richardson Extrapolation 395

11.4 Derivatives by Polynomial Interpolation 401

12 Numerical Integration 405

12.1 Introduction 405

12.2 Newton-Cotes Formulas 406

12.2.1 Rectangle Method 406

12.2.2 Midpoint Method 408

12.2.3 Trapezoidal Method 409

12.2.4 Simpson's Method 411

Simpson's 1/3 Method 411

Simpson's 3/8 Method 412

12.3 Romberg Integration 414

12.4 Gaussian Quadrature Methods 416

12.4.1 Gauss-Legendre Integration 417

12.4.2 Gauss-Hermite Integration 419

12.4.3 Gauss-Leguerre Integration 421

12.4.4 Gauss-Chebyshev Integration 423

12.5 Multiple Integration 424

12.6 Monte Carlo Methods 426

12.6.1 Monte Carlo Integration 427

12.6.2 The Metropolis Algorithm 428

12.7 Convolution Integrals 431

13 Statistical Functions 435

13.1 Introduction 435

13.2 Some Useful Tools 435

13.3 Basic Statistical Functions 438

13.3.1 Mean and Weighted Mean 438

13.3.2 Geometric and Weighted Geometric Mean 439

13.3.3 Harmonic and Weighted Harmonic Mean 440

13.3.4 Truncated Mean 441

13.3.5 Root Mean Square 441

13.3.6 Median, Range and Mode 442

13.3.7 Mean Deviation 444

13.3.8 Mean Deviation of the Mean 444

13.3.9 Mean Deviation of the Median 445

13.3.10 Variance and Standard Deviation 445

13.3.11 Moments About the Mean 447

13.3.12 Skewness 448

13.3.13 Kurtosis 449

13.3.14 Covariance and Correlation 451

13.3.15 Miscellaneous Utilities 453

13.3.16 Percentiles and Rank 456

14 Special Functions 461

14.1 Introduction 461

14.2 Factorials 461

14.3 Combinations and Permutations 464

14.3.1 Combinations 464

14.3.2 Permutations 467

14.4 Gamma Function 470

14.5 Beta Function 472

14.6 Error Function 472

14.7 Sine and Cosine Integral Functions 474

14.8 Laguerre Polynomials 475

14.9 Hermite Polynomials 476

14.10 Chebyshev Polynomials 477

14.11 Legendre Polynomials 479

14.12 Bessel Functions 480

15 Curve Fitting Methods 483

15.1 Introduction 483

15.2 Least Squares Fit 484

15.2.1 Straight-Line Fit 485

15.3 Weighted Least Squares Fit 488

15.3.1 Weighted Straight-Line Fit 488

15.4 Linear Regression 492

15.4.1 Polynomial Fit 496

15.4.2 Exponential Fit 497

15.5 The X(2) Test for Goodness of Fit 499

16 Ordinary Differential Equations 503

16.1 Introduction 503

16.2 Euler Method 505

16.3 Runge-Kutta Methods 506

16.3.1 Second-Order Runge-Kutta Method 507

16.3.2 Fourth-Order Runge-Kutta Method 508

16.3.3 Runge-Kutta-Fehlberg Method 510

16.4 Coupled Differential Equations 513

17 Partial Differential Equations 517

17.1 Introduction 517

17.2 The Finite Difference Method 520

17.3 Parabolic Partial Differential Equations 521

17.3.1 The Crank-Nicolson Method 525

17.4 Hyperbolic Partial Differential Equations 527

17.5 Elliptic Partial Differential Equations 532

18 Optimization Methods 539

18.1 Introduction 539

18.2 Gradient Descent Method 541

18.3 Linear Programming 544

18.3.1 The Revised Simplex Method 546

18.4 Simulated Annealing Method 550

18.5 Genetic Algorithms 555

References 571

Index 576