Numerical methods, algorithms and tools in C#

Although C, C++, Java, and Fortran are well-established programming languages, the relatively new C# is much easier to use for solving complex scientific and engineering problems. Numerical Methods, Algorithms and Tools in C# presents a broad collection of practical, ready-to-use mathematical routines employing the exciting, easy-to-learn C# programming language from Microsoft.

The book focuses on standard numerical methods, novel object-oriented techniques, and the latest Microsoft .NET programming environment. It covers complex number functions, data sorting and searching algorithms, bit manipulation, interpolation methods, numerical manipulation of linear algebraic equations, and numerical methods for calculating approximate solutions of non-linear equations. The author discusses alternative ways to obtain computer-generated pseudo-random numbers and real random numbers generated by naturally occurring physical phenomena. He also describes various methods for approximating integrals and special functions, routines for performing statistical analyses of data, and least squares and numerical curve fitting methods for analyzing experimental data, along with numerical methods for solving ordinary and partial differential equations. The final chapter offers optimization methods for the minimization or maximization of functions.

Exploiting the useful features of C#, this book shows how to write efficient, mathematically intense object-oriented computer programs. The vast array of practical examples presented can be easily customized and implemented to solve complex engineering and scientific problems typically found in real-world computer applications.

Author Notes

Waldemar Dos Passos is a computer programming consultant in the Silicon Valley area of California. After completing his undergraduate education at the University of California, Berkeley, Dr. Dos Passos earned an M.S. in computer science and engineering along with a Ph.D. in physics from the University of Michigan, Ann Arbor. With more than twenty years of computer programming experience, he has published several papers in physics journals.

l Introduction	p. 1
1.1 C# and the.NET Framework	p. 1
1.2 Installing C# and the.NET Framework	p. 3
1.3 Overview of Object-Oriented Programming (OOP)	p. 3
1.4 Your First C# Program	p. 4
1.5 Overview of the IDE Debugger	p. 9
1.6 Overview of the C# Language	p. 11
1.6.1 Data Types	p. 12
1.6.2 Value Types	p. 13
1.6.3 Reference Types	p. 14
1.6.4 Type-Parameter Types	p. 16
1.6.5 Pointer Types	p. 17
1.6.6 Variable Declaration	p. 17
1.6.7 Constant Declaration	p. 18
1.6.8 Nullable Types	p. 18
1.6.9 Scope	p. 18
1.6.10 Characters	p. 18
1.6.11 Strings	p. 19
1.6.12 Formatting of Output Data	p. 19
1.6.13 Type Conversion	p. 20
1.6.14 Reading Keyboard Input Data	p. 23
1.6.15 Basic Expressions and Operators	p. 24
1.6.16 Program Flow Mechanisms	p. 27
1.6.17 Jump Statements	p. 29
1.6.18 Arrays	p. 30
1.6.19 Enumerations	p. 32
1.6.20 Structures	p. 32
1.6.21 Exceptions	p. 33
1.6.22 Classes	p. 34
Constructors and Destructors	p. 37
Properties	p. 38
Methods	p. 38
1.6.23 Indexers	p. 42
1.6.24 Overloading Methods, Constructors and Operators	p. 42
1.6.25 Delegates	p. 43
1.6.26 Events	p. 46
1.6.27 Collections	p. 57
1.6.28 File Input/Output	p. 60
1.6.29 Output Reliability, Accuracy and Precision	p. 65
2 The.NET Framework Math Class Library	p. 73
2.1 Introduction	p. 73
2.2 The.NET Framework Math Class - Fields	p. 73
2.2.1 The Math. pi and Math. e Fields	p. 73
2.3 The.NET Framework Math Class - Methods	p. 74
2.3.1 The Minimum and Maximum Methods	p. 74
2.3.2 The Power, Exponential and Logarithmic Methods	p. 74
2.3.3 Special Multiplication, Division and Remainder Methods	p. 76
2.3.4 The Absolute Value Method	p. 77
2.3.5 The Sign Method	p. 78
2.3.6 Angular Units of Measurement	p. 78
2.3.7 The Trigonometric Functions	p. 81
2.3.8 The Inverse Trigonometric Functions	p. 82
2.3.9 The Hyperbolic Functions	p. 86
2.3.10 The Inverse Hyperbolic Functions	p. 88
2.3.11 Rounding Off Numeric Data	p. 89
The Ceiling Method	p. 89
The Floor Method	p. 90
The Truncation Method	p. 90
The Round Method	p. 91
3 Vectors and Matrices	p. 97
3.1 Introduction	p. 97
3.2 A Real Number Vector Library in C#	p. 98
3.3 A Real Number Matrix Library in C#	p. 106
4 Complex Numbers	p. 121
4.1 Introduction	p. 121
4.2 Fundamental Concepts	p. 121
4.3 Complex Number Arithmetic	p. 123
4.4 Elementary Functions of a Complex Number	p. 125
4.4.1 Exponentials	p. 125
4.4.2 Logarithms	p. 125
4.4.3 Powers and Roots	p. 127
4.4.4 Trigonometric and Hyperbolic Functions	p. 128
4.4.5 Inverse Trigonometric and Hyperbolic Functions	p. 130
4.5 A Complex Number Library in C#	p. 132
4.6 A Complex Number Vector Library in C#	p. 151
4.7 A Complex Number Matrix Library in C#	p. 158
4.8 Generic vs. Non-Generic Coding	p. 168
5 Sorting and Searching Algorithms	p. 171
5.1 Introduction	p. 171
5.2 Sorting Algorithms	p. 172
5.3 Comparison Sorts	p. 175
5.3.1 Bubble Sort	p. 175
5.3.2 Cocktail Sort	p. 178
5.3.3 Odd-Even Sort	p. 178
5.3.4 Comb Sort	p. 179
5.3.5 Gnome Sort	p. 180
5.3.6 Quicksort	p. 181
5.3.7 Insertion Sort	p. 182
5.3.8 Shell Sort	p. 183
5.3.9 Selection Sort	p. 184
5.3.10 Merge Sort	p. 185
5.3.11 Bucket Sort	p. 186
5.3.12 Heap Sort	p. 187
5.4 Count Sort	p. 188
5.5 Radix Sort	p. 189
5.6 Search Algorithms	p. 191
5.6.1 Linear Search	p. 192
5.6.2 Binary Search	p. 193
5.6.3 Interpolation Search	p. 193
5.6.4 Searching for the Maximum and Minimum Values	p. 194
5.6.5 Searching for the N-th Largest or M-th Smallest Value	p. 195
5.6.6 Some Useful Utilities	p. 196
6 Bits and Bytes	p. 199
6.1 Introduction	p. 199
6.2 Numeric Systems	p. 199
6.3 Bit Manipulation and Bitwise Operators	p. 202
6.4 Assorted Bits and Bytes	p. 223
7 Interpolation	p. 229
7.1 Introduction	p. 229
7.2 Linear Interpolation	p. 230
7.3 Bilinear Interpolation	p. 231
7.4 Polynomial Interpolation	p. 234
7.4.1 Lagrange Interpolation	p. 234
7.4.2 Barycentric Interpolation	p. 236
7.4.3 Newton's Divided Differences Interpolation	p. 238
7.5 Cubic Spline Interpolation	p. 242
7.5.1 Natural Cubic Splines	p. 244
7.5.2 Clamped Cubic Splines	p. 247
8 Linear Equations	p. 251
8.1 Introduction	p. 251
8.2 Gaussian Elimination	p. 253
8.3 Gauss-Jordan Elimination	p. 254
8.4 LU Decomposition	p. 256
8.5 Iteration Methods	p. 259
8.5.1 Gauss-Jacobi Iteration	p. 259
8.5.2 Gauss-Seidel Iteration	p. 261
8.6 Eigenvalues and Jacobi's Algorithm	p. 264
9 Nonlinear Equations	p. 271
9.1 Introduction	p. 271
9.2 Linear Incremental Method	p. 272
9.3 Bisection Method	p. 274
9.4 The Secant Method	p. 276
9.5 False Positioning Method	p. 277
9.6 Fixed Point Iteration	p. 279
9.7 Newton-Raphson Method	p. 280
10 Random Numbers	p. 283
10.1 Introduction	p. 283
10.2 The C# Built-In Random Number Generator	p. 284
10.3 Other Random Number Generators	p. 290
10.4 True Random Number Generators	p. 295
10.5 Random Variate Generation Methods	p. 299
10.6 Histograms	p. 309
10.7 Random Variate Generation	p. 312
10.7.1 Discrete Distributions	p. 312
Bernoulli Distribution	p. 312
Binoulli Distribution	p. 315
Geometric Distribution	p. 317
Negative Binomial Distribution	p. 320
Poisson Distribution	p. 322
Uniform Distribution (discrete)	p. 326
10.7.2 Continuous Distributions	p. 328
Beta Distribution	p. 328
Beta Prime Distribution	p. 330
Cauchy Distribution	p. 332
Chi Distribution	p. 334
Chi-Square Distribution	p. 337
Erlang Distribution	p. 340
Exponential Distribution	p. 343
Extreme Value Distribution	p. 345
Gamma Distribution	p. 347
Laplace Distribution	p. 349
Logistic Distribution	p. 352
Lognormal Distribution	p. 354
Normal Distribution	p. 356
Pareto Distribution	p. 359
Rayleigh Distribution	p. 361
Student-t Distribution	p. 363
Triangular Distribution	p. 365
Uniform Distribution (continuous)	p. 368
Weibull Distribution	p. 370
10.8 Shuffling Algorithms	p. 372
10.9 Adding Random Noise to Data	p. 376
10.10 Removing Random Noise from Data	p. 379
11 Numerical Differentiation	p. 383
11.1 Introduction	p. 383
11.2 Finite Difference Formulas	p. 383
11.2.1 Forward Difference Method	p. 385
11.2.2 Backward Difference Method	p. 387
11.2.3 Central Difference Method	p. 390
11.2.4 Improved Central Difference Method	p. 392
11.3 Richardson Extrapolation	p. 395
11.4 Derivatives by Polynomial Interpolation	p. 401
12 Numerical Integration	p. 405
12.1 Introduction	p. 405
12.2 Newton-Cotes Formulas	p. 406
12.2.1 Rectangle Method	p. 406
12.2.2 Midpoint Method	p. 408
12.2.3 Trapezoidal Method	p. 409
12.2.4 Simpson's Method	p. 411
Simpson's 1/3 Method	p. 411
Simpson's 3/8 Method	p. 412
12.3 Romberg Integration	p. 414
12.4 Gaussian Quadrature Methods	p. 416
12.4.1 Gauss-Legendre Integration	p. 417
12.4.2 Gauss-Hermite Integration	p. 419
12.4.3 Gauss-Leguerre Integration	p. 421
12.4.4 Gauss-Chebyshev Integration	p. 423
12.5 Multiple Integration	p. 424
12.6 Monte Carlo Methods	p. 426
12.6.1 Monte Carlo Integration	p. 427
12.6.2 The Metropolis Algorithm	p. 428
12.7 Convolution Integrals	p. 431
13 Statistical Functions	p. 435
13.1 Introduction	p. 435
13.2 Some Useful Tools	p. 435
13.3 Basic Statistical Functions	p. 438
13.3.1 Mean and Weighted Mean	p. 438
13.3.2 Geometric and Weighted Geometric Mean	p. 439
13.3.3 Harmonic and Weighted Harmonic Mean	p. 440
13.3.4 Truncated Mean	p. 441
13.3.5 Root Mean Square	p. 441
13.3.6 Median, Range and Mode	p. 442
13.3.7 Mean Deviation	p. 444
13.3.8 Mean Deviation of the Mean	p. 444
13.3.9 Mean Deviation of the Median	p. 445
13.3.10 Variance and Standard Deviation	p. 445
13.3.11 Moments About the Mean	p. 447
13.3.12 Skewness	p. 448
13.3.13 Kurtosis	p. 449
13.3.14 Covariance and Correlation	p. 451
13.3.15 Miscellaneous Utilities	p. 453
13.3.16 Percentiles and Rank	p. 456
14 Special Functions	p. 461
14.1 Introduction	p. 461
14.2 Factorials	p. 461
14.3 Combinations and Permutations	p. 464
14.3.1 Combinations	p. 464
14.3.2 Permutations	p. 467
14.4 Gamma Function	p. 470
14.5 Beta Function	p. 472
14.6 Error Function	p. 472
14.7 Sine and Cosine Integral Functions	p. 474
14.8 Laguerre Polynomials	p. 475
14.9 Hermite Polynomials	p. 476
14.10 Chebyshev Polynomials	p. 477
14.11 Legendre Polynomials	p. 479
14.12 Bessel Functions	p. 480
15 Curve Fitting Methods	p. 483
15.1 Introduction	p. 483
15.2 Least Squares Fit	p. 484
15.2.1 Straight-Line Fit	p. 485
15.3 Weighted Least Squares Fit	p. 488
15.3.1 Weighted Straight-Line Fit	p. 488
15.4 Linear Regression	p. 492
15.4.1 Polynomial Fit	p. 496
15.4.2 Exponential Fit	p. 497
15.5 The X(2) Test for Goodness of Fit	p. 499
16 Ordinary Differential Equations	p. 503
16.1 Introduction	p. 503
16.2 Euler Method	p. 505
16.3 Runge-Kutta Methods	p. 506
16.3.1 Second-Order Runge-Kutta Method	p. 507
16.3.2 Fourth-Order Runge-Kutta Method	p. 508
16.3.3 Runge-Kutta-Fehlberg Method	p. 510
16.4 Coupled Differential Equations	p. 513
17 Partial Differential Equations	p. 517
17.1 Introduction	p. 517
17.2 The Finite Difference Method	p. 520
17.3 Parabolic Partial Differential Equations	p. 521
17.3.1 The Crank-Nicolson Method	p. 525
17.4 Hyperbolic Partial Differential Equations	p. 527
17.5 Elliptic Partial Differential Equations	p. 532
18 Optimization Methods	p. 539
18.1 Introduction	p. 539
18.2 Gradient Descent Method	p. 541
18.3 Linear Programming	p. 544
18.3.1 The Revised Simplex Method	p. 546
18.4 Simulated Annealing Method	p. 550
18.5 Genetic Algorithms	p. 555
References	p. 571
Index	p. 576

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