Numerical methods, algorithms, and tools in C♯

__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. Title Page......Page 3 Front Cover......Page 1 Copyright and ISBN......Page 4 Preface......Page 5 Acknowledgements......Page 8 Contents......Page 11 1.1 C# and the .NET Framework......Page 18 1.3 Overview of Object-Oriented Programming (OOP)......Page 20 1.4 Your First C# Program......Page 21 1.5 Overview of the IDE Debugger......Page 26 1.6 Overview of the C# Language......Page 28 1.6.1 Data Types......Page 29 1.6.2 Value Types......Page 30 1.6.3 Reference Types......Page 31 1.6.4 Type-Parameter Types......Page 33 1.6.6 Variable Declaration......Page 34 1.6.10 Characters......Page 35 1.6.12 Formatting of Output Data......Page 36 1.6.13 Type Conversion......Page 37 1.6.14 Reading Keyboard Input Data......Page 40 1.6.15 Basic Expressions and Operators......Page 41 Selection Statements......Page 44 Loop Sequences......Page 45 1.6.17 Jump Statements......Page 46 1.6.18 Arrays......Page 47 1.6.20 Structures......Page 49 1.6.21 Exceptions......Page 50 1.6.22 Classes......Page 51 Constructors and Destructors......Page 54 Methods......Page 55 1.6.24 OverloadingMethods, Constructors and Operators......Page 59 1.6.25 Delegates......Page 60 1.6.26 Events......Page 63 1.6.27 Collections......Page 74 1.6.28 File Input/Output......Page 77 1.6.29 Output Reliability, Accuracy and Precision......Page 82 2.2.1 The Math.PI and Math.E Fields......Page 90 2.3.2 The Power, Exponential and Logarithmic Methods......Page 91 2.3.3 Special Multiplication, Division and Remainder Methods......Page 93 2.3.4 The Absolute Value Method......Page 94 2.3.6 Angular Units of Measurement......Page 95 2.3.7 The Trigonometric Functions......Page 98 2.3.8 The Inverse Trigonometric Functions......Page 99 2.3.9 The Hyperbolic Functions......Page 103 2.3.10 The Inverse Hyperbolic Functions......Page 105 The Ceiling Method......Page 106 The Truncation Method......Page 107 The Round Method......Page 108 3.1 Introduction......Page 114 3.2 A Real Number Vector Library in C#......Page 115 3.3 A Real Number Matrix Library in C#......Page 123 4.2 Fundamental Concepts......Page 138 4.3 Complex Number Arithmetic......Page 140 4.4.2 Logarithms......Page 142 4.4.3 Powers and Roots......Page 144 4.4.4 Trigonometric and Hyperbolic Functions......Page 145 4.4.5 Inverse Trigonometric and Hyperbolic Functions......Page 147 4.5 A Complex Number Library in C#......Page 149 4.6 A Complex Number Vector Library in C#......Page 168 4.7 A Complex Number Matrix Library in C#......Page 175 4.8 Generic vs. Non-Generic Coding......Page 185 5.1 Introduction......Page 188 5.2 Sorting Algorithms......Page 189 5.3.1 Bubble Sort......Page 192 5.3.3 Odd-Even Sort......Page 195 5.3.4 Comb Sort......Page 196 5.3.5 Gnome Sort......Page 197 5.3.6 Quicksort......Page 198 5.3.7 Insertion Sort......Page 199 5.3.8 Shell Sort......Page 200 5.3.9 Selection Sort......Page 201 5.3.10 Merge Sort......Page 202 5.3.11 Bucket Sort......Page 203 5.3.12 Heap Sort......Page 204 5.4 Count Sort......Page 205 5.5 Radix Sort......Page 206 5.6 Search Algorithms......Page 208 5.6.1 Linear Search......Page 209 5.6.3 Interpolation Search......Page 210 5.6.4 Searching for the Maximum and Minimum Values......Page 211 5.6.5 Searching for the N-th Largest or M-th Smallest Value......Page 212 5.6.6 Some Useful Utilities......Page 213 6.2 Numeric Systems......Page 216 6.3 Bit Manipulation and Bitwise Operators......Page 219 6.4 Assorted Bits and Bytes......Page 240 7.1 Introduction......Page 246 7.2 Linear Interpolation......Page 247 7.3 Bilinear Interpolation......Page 248 7.4.1 Lagrange Interpolation......Page 251 7.4.2 Barycentric Interpolation......Page 253 7.4.3 Newton’s Divided Differences Interpolation......Page 255 7.5 Cubic Spline Interpolation......Page 259 7.5.1 Natural Cubic Splines......Page 261 7.5.2 Clamped Cubic Splines......Page 264 8.1 Introduction......Page 268 8.2 Gaussian Elimination......Page 270 8.3 Gauss-Jordan Elimination......Page 271 8.4 LU Decomposition......Page 273 8.5.1 Gauss-Jacobi Iteration......Page 276 8.5.2 Gauss-Seidel Iteration......Page 278 8.6 Eigenvalues and Jacobi’s Algorithm......Page 281 9.1 Introduction......Page 288 9.2 Linear Incremental Method......Page 289 9.3 Bisection Method......Page 291 9.4 The Secant Method......Page 293 9.5 False Positioning Method......Page 294 9.6 Fixed Point Iteration......Page 296 9.7 Newton-Raphson Method......Page 297 10.1 Introduction......Page 300 10.2 The C# Built-In Random Number Generator......Page 301 10.3 Other Random Number Generators......Page 307 10.4 True Random Number Generators......Page 312 10.5 Random Variate Generation Methods......Page 316 10.6 Histograms......Page 326 Bernoulli Distribution......Page 329 Binomial Distribution......Page 332 Geometric Distribution......Page 334 Negative Binomial Distribution......Page 337 Poisson Distribution......Page 339 Uniform Distribution (discrete)......Page 343 Beta Distribution......Page 345 Beta Prime Distribution......Page 347 Cauchy Distribution......Page 349 Chi Distribution......Page 351 Chi-Square Distribution......Page 354 Erlang Distribution......Page 357 Exponential Distribution......Page 360 Extreme Value Distribution......Page 362 Gamma Distribution......Page 364 Laplace Distribution......Page 366 Logistic Distribution......Page 369 Lognormal Distribution......Page 371 Normal Distribution......Page 373 Pareto Distribution......Page 376 Rayleigh Distribution......Page 378 Student-t Distribution......Page 380 Triangular Distribution......Page 382 Uniform Distribution (continuous)......Page 385 Weibull Distribution......Page 387 10.8 Shuffling Algorithms......Page 389 10.9 Adding Random Noise to Data......Page 393 10.10 Removing Random Noise from Data......Page 396 11.2 Finite Difference Formulas......Page 400 11.2.1 Forward Difference Method......Page 402 11.2.2 Backward DifferenceMethod......Page 404 11.2.3 Central Difference Method......Page 407 11.2.4 Improved Central Difference Method......Page 409 11.3 Richardson Extrapolation......Page 412 11.4 Derivatives by Polynomial Interpolation......Page 418 12.1 Introduction......Page 422 12.2.1 Rectangle Method......Page 423 12.2.2 Midpoint Method......Page 425 12.2.3 Trapezoidal Method......Page 426 Simpson’s 1/3 Method......Page 428 Simpson’s 3/8 Method......Page 429 12.3 Romberg Integration......Page 431 12.4 Gaussian Quadrature Methods......Page 433 12.4.1 Gauss-Legendre Integration......Page 434 12.4.2 Gauss-Hermite Integration......Page 436 12.4.3 Gauss-Leguerre Integration......Page 438 12.4.4 Gauss-Chebyshev Integration......Page 440 12.5 Multiple Integration......Page 441 12.6 Monte Carlo Methods......Page 443 12.6.1 Monte Carlo Integration......Page 444 12.6.2 The Metropolis Algorithm......Page 445 12.7 Convolution Integrals......Page 448 13.2 Some Useful Tools......Page 452 13.3.1 Mean and Weighted Mean......Page 455 13.3.2 Geometric and Weighted Geometric Mean......Page 456 13.3.3 Harmonic and Weighted Harmonic Mean......Page 457 13.3.5 Root Mean Square......Page 458 13.3.6 Median, Range and Mode......Page 459 13.3.8 Mean Deviation of the Mean......Page 461 13.3.10 Variance and Standard Deviation......Page 462 13.3.11 Moments About the Mean......Page 464 13.3.12 Skewness......Page 465 13.3.13 Kurtosis......Page 466 13.3.14 Covariance and Correlation......Page 468 13.3.15 Miscellaneous Utilities......Page 470 13.3.16 Percentiles and Rank......Page 473 14.2 Factorials......Page 478 14.3.1 Combinations......Page 481 14.3.2 Permutations......Page 484 14.4 Gamma Function......Page 487 14.6 Error Function......Page 489 14.7 Sine and Cosine Integral Functions......Page 491 14.8 Laguerre Polynomials......Page 492 14.9 Hermite Polynomials......Page 493 14.10 Chebyshev Polynomials......Page 494 14.11 Legendre Polynomials......Page 496 14.12 Bessel Functions......Page 497 15.1 Introduction......Page 500 15.2 Least Squares Fit......Page 501 15.2.1 Straight-Line Fit......Page 502 15.3.1 Weighted Straight-Line Fit......Page 505 15.4 Linear Regression......Page 509 15.4.1 Polynomial Fit......Page 513 15.4.2 Exponential Fit......Page 514 15.5 The X.........Page 516 16.1 Introduction......Page 520 16.2 Euler Method......Page 522 16.3 Runge-Kutta Methods......Page 523 16.3.1 Second-Order Runge-Kutta Method......Page 524 16.3.2 Fourth-Order Runge-Kutta Method......Page 525 16.3.3 Runge-Kutta-FehlbergMethod......Page 527 16.4 Coupled Differential Equations......Page 530 17.1 Introduction......Page 534 17.2 The Finite Difference Method......Page 537 17.3 Parabolic Partial Differential Equations......Page 538 17.3.1 The Crank-Nicolson Method......Page 542 17.4 Hyperbolic Partial Differential Equations......Page 544 17.5 Elliptic Partial Differential Equations......Page 549 18.1 Introduction......Page 556 18.2 Gradient Descent Method......Page 558 18.3 Linear Programming......Page 561 18.3.1 The Revised Simplex Method......Page 563 18.4 Simulated Annealing Method......Page 567 18.5 Genetic Algorithms......Page 572 References......Page 588 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. Along with providing the C♯ source code online, this book presents practical, ready-to-use mathematical routines employing the C♯ programming language from Microsoft. It shows how to write mathematically intense object-oriented computer programs. It covers a spectrum of computational tools, including sorting algorithms and optimization methods

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.