Numerical Methods, Algorithms and Tools in C'

✍ Scribed by Waldemar Dos Passos

• Publisher: CRC
• Year: 2009
• Tongue: English
• Leaves: 583
• Category: Library

✦ Synopsis

Comprehensive Coverage of the New, Easy-to-Learn 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.

✦ Table of Contents

Cover
......Page 1
Numerical Methods, Algorithms and Tools in C#......Page 3
Preface......Page 5
Acknowledgements......Page 8
Contents......Page 10
1.1 C# and the .NET Framework......Page 17
1.3 Overview of Object-Oriented Programming (OOP)......Page 19
1.4 Your First C# Program......Page 20
1.5 Overview of the IDE Debugger......Page 25
1.6 Overview of the C# Language......Page 27
1.6.1 Data Types......Page 28
1.6.2 Value Types......Page 29
1.6.3 Reference Types......Page 30
1.6.4 Type-Parameter Types......Page 32
1.6.6 Variable Declaration......Page 33
1.6.10 Characters......Page 34
1.6.12 Formatting of Output Data......Page 35
1.6.13 Type Conversion......Page 36
1.6.14 Reading Keyboard Input Data......Page 39
1.6.15 Basic Expressions and Operators......Page 40
Selection Statements......Page 43
Loop Sequences......Page 44
1.6.17 Jump Statements......Page 45
1.6.18 Arrays......Page 46
1.6.20 Structures......Page 48
1.6.21 Exceptions......Page 49
1.6.22 Classes......Page 50
Constructors and Destructors......Page 53
Methods......Page 54
1.6.24 OverloadingMethods, Constructors and Operators......Page 58
1.6.25 Delegates......Page 59
1.6.26 Events......Page 62
1.6.27 Collections......Page 73
1.6.28 File Input/Output......Page 76
1.6.29 Output Reliability, Accuracy and Precision......Page 81
2.2.1 The Math.PI and Math.E Fields......Page 89
2.3.2 The Power, Exponential and Logarithmic Methods......Page 90
2.3.3 Special Multiplication, Division and Remainder Methods......Page 92
2.3.4 The Absolute Value Method......Page 93
2.3.6 Angular Units of Measurement......Page 94
2.3.7 The Trigonometric Functions......Page 97
2.3.8 The Inverse Trigonometric Functions......Page 98
2.3.9 The Hyperbolic Functions......Page 102
2.3.10 The Inverse Hyperbolic Functions......Page 104
The Ceiling Method......Page 105
The Truncation Method......Page 106
The Round Method......Page 107
3.1 Introduction......Page 112
3.2 A Real Number Vector Library in C#......Page 113
3.3 A Real Number Matrix Library in C#......Page 121
4.2 Fundamental Concepts......Page 136
4.3 Complex Number Arithmetic......Page 138
4.4.2 Logarithms......Page 140
4.4.3 Powers and Roots......Page 142
4.4.4 Trigonometric and Hyperbolic Functions......Page 143
4.4.5 Inverse Trigonometric and Hyperbolic Functions......Page 145
4.5 A Complex Number Library in C#......Page 147
4.6 A Complex Number Vector Library in C#......Page 166
4.7 A Complex Number Matrix Library in C#......Page 173
4.8 Generic vs. Non-Generic Coding......Page 183
5.1 Introduction......Page 186
5.2 Sorting Algorithms......Page 187
5.3.1 Bubble Sort......Page 190
5.3.3 Odd-Even Sort......Page 193
5.3.4 Comb Sort......Page 194
5.3.5 Gnome Sort......Page 195
5.3.6 Quicksort......Page 196
5.3.7 Insertion Sort......Page 197
5.3.8 Shell Sort......Page 198
5.3.9 Selection Sort......Page 199
5.3.10 Merge Sort......Page 200
5.3.11 Bucket Sort......Page 201
5.3.12 Heap Sort......Page 202
5.4 Count Sort......Page 203
5.5 Radix Sort......Page 204
5.6 Search Algorithms......Page 206
5.6.1 Linear Search......Page 207
5.6.3 Interpolation Search......Page 208
5.6.4 Searching for the Maximum and Minimum Values......Page 209
5.6.5 Searching for the N-th Largest or M-th Smallest Value......Page 210
5.6.6 Some Useful Utilities......Page 211
6.2 Numeric Systems......Page 213
6.3 Bit Manipulation and Bitwise Operators......Page 216
6.4 Assorted Bits and Bytes......Page 237
7.1 Introduction......Page 243
7.2 Linear Interpolation......Page 244
7.3 Bilinear Interpolation......Page 245
7.4.1 Lagrange Interpolation......Page 248
7.4.2 Barycentric Interpolation......Page 250
7.4.3 Newton’s Divided Differences Interpolation......Page 252
7.5 Cubic Spline Interpolation......Page 256
7.5.1 Natural Cubic Splines......Page 258
7.5.2 Clamped Cubic Splines......Page 261
8.1 Introduction......Page 265
8.2 Gaussian Elimination......Page 267
8.3 Gauss-Jordan Elimination......Page 268
8.4 LU Decomposition......Page 270
8.5.1 Gauss-Jacobi Iteration......Page 273
8.5.2 Gauss-Seidel Iteration......Page 275
8.6 Eigenvalues and Jacobi’s Algorithm......Page 278
9.1 Introduction......Page 285
9.2 Linear Incremental Method......Page 286
9.3 Bisection Method......Page 288
9.4 The Secant Method......Page 290
9.5 False Positioning Method......Page 291
9.6 Fixed Point Iteration......Page 293
9.7 Newton-Raphson Method......Page 294
10.1 Introduction......Page 296
10.2 The C# Built-In Random Number Generator......Page 297
10.3 Other Random Number Generators......Page 303
10.4 True Random Number Generators......Page 308
10.5 Random Variate Generation Methods......Page 312
10.6 Histograms......Page 322
Bernoulli Distribution......Page 325
Binomial Distribution......Page 328
Geometric Distribution......Page 330
Negative Binomial Distribution......Page 333
Poisson Distribution......Page 335
Uniform Distribution (discrete)......Page 339
Beta Distribution......Page 341
Beta Prime Distribution......Page 343
Cauchy Distribution......Page 345
Chi Distribution......Page 347
Chi-Square Distribution......Page 350
Erlang Distribution......Page 353
Exponential Distribution......Page 356
Extreme Value Distribution......Page 358
Gamma Distribution......Page 360
Laplace Distribution......Page 362
Logistic Distribution......Page 365
Lognormal Distribution......Page 367
Normal Distribution......Page 369
Pareto Distribution......Page 372
Rayleigh Distribution......Page 374
Student-t Distribution......Page 376
Triangular Distribution......Page 378
Uniform Distribution (continuous)......Page 381
Weibull Distribution......Page 383
10.8 Shuffling Algorithms......Page 385
10.9 Adding Random Noise to Data......Page 389
10.10 Removing Random Noise from Data......Page 392
11.2 Finite Difference Formulas......Page 395
11.2.1 Forward Difference Method......Page 397
11.2.2 Backward DifferenceMethod......Page 399
11.2.3 Central Difference Method......Page 402
11.2.4 Improved Central Difference Method......Page 404
11.3 Richardson Extrapolation......Page 407
11.4 Derivatives by Polynomial Interpolation......Page 413
12.1 Introduction......Page 416
12.2.1 Rectangle Method......Page 417
12.2.2 Midpoint Method......Page 419
12.2.3 Trapezoidal Method......Page 420
Simpson’s 1/3 Method......Page 422
Simpson’s 3/8 Method......Page 423
12.3 Romberg Integration......Page 425
12.4 Gaussian Quadrature Methods......Page 427
12.4.1 Gauss-Legendre Integration......Page 428
12.4.2 Gauss-Hermite Integration......Page 430
12.4.3 Gauss-Leguerre Integration......Page 432
12.4.4 Gauss-Chebyshev Integration......Page 434
12.5 Multiple Integration......Page 435
12.6 Monte Carlo Methods......Page 437
12.6.1 Monte Carlo Integration......Page 438
12.6.2 The Metropolis Algorithm......Page 439
12.7 Convolution Integrals......Page 442
13.2 Some Useful Tools......Page 446
13.3.1 Mean and Weighted Mean......Page 449
13.3.2 Geometric and Weighted Geometric Mean......Page 450
13.3.3 Harmonic and Weighted Harmonic Mean......Page 451
13.3.5 Root Mean Square......Page 452
13.3.6 Median, Range and Mode......Page 453
13.3.8 Mean Deviation of the Mean......Page 455
13.3.10 Variance and Standard Deviation......Page 456
13.3.11 Moments About the Mean......Page 458
13.3.12 Skewness......Page 459
13.3.13 Kurtosis......Page 460
13.3.14 Covariance and Correlation......Page 462
13.3.15 Miscellaneous Utilities......Page 464
13.3.16 Percentiles and Rank......Page 467
14.2 Factorials......Page 472
14.3.1 Combinations......Page 475
14.3.2 Permutations......Page 478
14.4 Gamma Function......Page 481
14.6 Error Function......Page 483
14.7 Sine and Cosine Integral Functions......Page 485
14.8 Laguerre Polynomials......Page 486
14.9 Hermite Polynomials......Page 487
14.10 Chebyshev Polynomials......Page 488
14.11 Legendre Polynomials......Page 490
14.12 Bessel Functions......Page 491
15.1 Introduction......Page 493
15.2 Least Squares Fit......Page 494
15.2.1 Straight-Line Fit......Page 495
15.3.1 Weighted Straight-Line Fit......Page 498
15.4 Linear Regression......Page 502
15.4.1 Polynomial Fit......Page 506
15.4.2 Exponential Fit......Page 507
15.5 The X…......Page 509
16.1 Introduction......Page 513
16.2 Euler Method......Page 515
16.3 Runge-Kutta Methods......Page 516
16.3.1 Second-Order Runge-Kutta Method......Page 517
16.3.2 Fourth-Order Runge-Kutta Method......Page 518
16.3.3 Runge-Kutta-FehlbergMethod......Page 520
16.4 Coupled Differential Equations......Page 523
17.1 Introduction......Page 526
17.2 The Finite Difference Method......Page 529
17.3 Parabolic Partial Differential Equations......Page 530
17.3.1 The Crank-Nicolson Method......Page 534
17.4 Hyperbolic Partial Differential Equations......Page 536
17.5 Elliptic Partial Differential Equations......Page 541
18.1 Introduction......Page 548
18.2 Gradient Descent Method......Page 550
18.3 Linear Programming......Page 553
18.3.1 The Revised Simplex Method......Page 555
18.4 Simulated Annealing Method......Page 559
18.5 Genetic Algorithms......Page 564
References......Page 579

✦ Subjects

Библиотека;Компьютерная литература;C#;