About this Book

This book provides an in-depth exposition of the various numerical methods used in real-world scientific and engineering computations. It emphasizes the practical aspects of C# numerical methods and mathematical functions programming, and discusses in detail various techniques that will enable you to implement these numerical methods in your own .NET applications.

Ideal for scientists, engineers, and students who would like to become more adept with numerical methods, Practical Numerical Methods with C# familiarizes you with:

• The basics of C# programming.
• The mathematical background and fundamentals of numerical methods.
• Math libraries for complex numbers and functions, real and complex vector and matrix operations, and special functions.
• Numerical methods for generating random numbers and random distribution functions.
• Various numerical methods for solving linear and nonlinear equations.
• Numerical differentiation and integration.
• Interpolations and curve fitting.
• Optimization of single-variable and multi-variable functions using a variety of techniques, including advanced simulated annealing and evolutionary algorithms.
• Numerical techniques for solving ordinary differential equations.
• Numerical methods for solving boundary value problems.
• Eigenvalue problems.

In addition, this book provides code examples for every math function and numerical method to show you how to use these functions and methods in your own .NET applications.

Table of Contents

Introduction
   Overview
   What This Book Includes
   Is This Book for You
   What Do You Need to Use This Book
   How The Book Is Organized
   Using Code Examples
   Customer Support

Chapter 1: Overview of C# Programming
   Your First C# program
   Data Types
      Value-Type Variables
      Reference-Type Variables
   Math Operators and Functions
      Math Operators
      Logical Operators
      Built-In Math Functions
      Math Operation Example
   Selection Statements
      If Statement
      Switch Statement
   Loop Statements
      For Loop
      Foreach Loop
      While Loop
      Do Loop
   Methods
      Creating a Method
      Calling a Method
   Properties
      Field Members
      Exposing a Property
      Using Properties
      Read-Only and Write-Only Properties
      Auto-Implemented properties

Chapter 2: Complex Numbers and Functions
   Complex Numbers and Operators
      Complex Numbers in C#
         Common Public Properties
         Test Public Properties
         Common Public Methods
         Equality and Hashing
      Complex Operators
         Unary Operator
         Addition Operator
         Subtraction Operator
         Multiplication Operator
         Division Operator
         Testing Math Operator
   Complex Functions
      Basic Math Functions
         Square Root Function
         Exponential Function
         Pow Function
         Logarithm Function
         Testing Math Functions
      Trigonometric Functions
         Sine Function
         Cosine Function
         Tangent Function
         Testing Trigonometrix Functions
      Inverse Trigonometric Functions
         Inverse Sine Function
         Inverse Cosine Function
         Inverse Tangent Function
         Testing Inverse Trigonometrix Functions
      Hyperbolic Trigonometric Functions
         Hyperbolic Sine Function
         Hyperbolic Cosine Function
         Hyperbolic Tangent Function
         Testing Hyperbolic Trigonometrix Functions
      Inverse Hyperbolic Trigonometric Functions
         Inverse Hyperbolic Sine Function
         Inverse Hyperbolic Cosine Function
         Inverse Hyperbolic Tangent Function
         Testing Hyperbolic Inverse Trigonometrix Functions

Chapter 3: Vectors and Matrices
   Real Vector Structure
      basic Definitions
      Mathematical Operators
      Vector Multiplications
         Scalar or Dot product
         Cross Product
         Triple Scalar Product
         Triple Vector Product
   Complex Vector Class
      Implementation
      Testing the VectorC Class
   Real Matrix Class
      basic Definitions
      Matrix and Vector Multiplications
         Vector from Matrix
         Swap, Transpose, and Trace
         Transformations
         Matrix Multiplication
      Mathematical Operators
      Determinant
      Inverse
      Testing the Real Matrix
   Complex matrix Class
      Implementation
      Testing the Complex Matrix

Chapter 4: Linear Algebraic Equations
   Gauss-Jordan Elimination
      Algorithms
      Implementation
      Testing Gauss-Jordan Elimination
   LU Decomposition
      Algorithms
      Implementation
      Matrix Inverse
      Testing LU Decomposition
   Iteration Methods
      Gauss-Jacobi Iteration
      Gauss-Seidel Iteration
      Testing Iteration Methods

Chapter 5: Nonlinear Equations
   Incremental Method
      Implementation
      Testing the Incremental Method
   Fixed Point Method
      Implementation
      Testing the Fixed Method
   Bisection Method
      Implementation
      Testing the Bisection Method
   False Position Method
      Implementation
      Testing the False Position Method
   Newton-Raphson Method
      Implementation
      Testing the Newton-Raphson Method
   Secant Method
      Implementation
      Testing the Secant Method
   Newton Multiroot Method
      Implementation
      Testing the Newton Multiroot Method
   Birge-Vieta Method
      Implementation
      Testing the Birge-Vieta Method
   System of Equations
      Algorithm
      Implementation
      Testing the NewtonMultiEquations Method

Chapter 6: Special Functions
   Gamma and Beta Functions
      Gamma Function
         Implementation
         testing the Gamma Method
      Beta Function
         Implementation
         testing the Beta Method
   Error Function
      Implementation
      Testing the Erf and Erfc Methods
   Sine and Cosine Integral Function
      Implementation
      Testing the Si and Ci Methods
   Laguerre Function
      Implementation
      Testing the Laguerre Method
   Hermite Polynomials
      Implementation
      Testing the Hermite Method
   Chebyshev Polynomials
      Implementation
      Testing the Chebyshev Method
   Legendre Polynomials
      Implementation
      Testing the Legendre Method
   Bessel Functions
      Implementation
      Testing the Bessel Method

Chapter 7: Random Numbers and Distribution Functions
   Built-in Random Number Generators
   Normal Distribution
      Probability Density Function
      Normal Random Number Generator
      Testing the Normal Distribution
   Exponential Distribution
      Probability Density Function
      Exponential Random Number Generator
      Testing the Exponential Distribution
   Chi and Chi-Square Distributions
      Probability Density Function
      Chi and Chi-Square Random Number Generators
      Testing the Chi and Chi-Square Distributions
   Cauchy Distribution
      Probability Density Function
      Cauchy Random Number Generator
      Testing the Cauchy Distribution
   Student T Distribution
      Probability Density Function
      Student T Random Number Generator
      Testing the Student T Distribution
   Gamma Distribution
      Probability Density Function
      Gamma Random Number Generator
      Testing the Gamma Distribution
   Beta Distribution
      Probability Density Function
      Beta Random Number Generator
      Testing the Beta Distribution
   Poisson Distribution
      Probability Density Function
      Poisson Random Number Generator
      Testing the Poisson Distribution
   Binomial Distribution
      Probability Density Function
      Binomial Random Number Generator
      Testing the Binomial Distribution

Chapter 8: Interpolation
   Linear Interpolation
      Algorithm
      Implementation
      Testing the Linear Interpolation
   Lagrange Interpolation
      Algorithm
      Implementation
      Testing the Lagrange Interpolation
   B Interpolationarycentric
      Algorithm
      Implementation
      Testing the Barycentric Interpolation
   Newton Divided Difference Interpolation
      Algorithm
      Implementation
      Testing the Newton Divided Difference Interpolation
   Cubic Spline Interpolation
      Algorithm
      Implementation
      Testing the Spline Interpolation
   Bilinear Interpolation
      Algorithm
      Implementation
      Testing the Bilinear Interpolation

Chapter 9: Curve Fitting
   Least Squares Fit
   Straight Line Fit
      Implementation
      Testing the Straight Line Fit
   Linear Regression
      Implementation
      Testing the Linear Regression
   Polynomial Fit
      Implementation
      Testing the Polynomial Fit
   Weighted Linear Regression
      Implementation
      Exponential Function Fit
   Simple Moving Average
      Implementation
      Testing the Simple Moving Average
   Weighted Moving Average
      Implementation
      Testing the Weighted Moving Average
   Exponential Moving Average
      Implementation
      Testing the Exponential Moving Average

Chapter 10: Optimization
   Bisection Method
      Implementation
      Testing the Bisection Method
   Golden Search Method
      Implementation
      Testing the Golden Search Method
   Newton Method
      Implementation
      Testing the Newton Method
   Brent Method
      Implementation
      Testing the Brent Method
   Newton Method for Multi-variable Functions
      Implementation
      Testing the MultiNewton Method
   Simplex Method
      Implementation
      Testing the Simplex Method
   Simulated Annealing Method
      Implementation
      Testing the Simulated Annealing Method
   Differential Evolution
      Algorithm
      Implementation
      Testing Differential Evolution

Chapter 11: Numerical Differentiation
   Finite Difference Formulas
   Forward Difference Method
      Implementation
      Testing the Forworward Difference Method
   Backward Difference Method
      Implementation
      Testing the Backworward Difference Method
   Central Difference Method
      Implementation
      Testing the Central Difference Method
   Extended Central Difference Method
      Implementation
      Testing the Extended central Difference Method
   Richardson Extrapolation
      Implementation
      Testing the Richardson Extrapolation
   Derivatives by Interpolation
      Implementation
      Testing the Derivatives by Interpolation

Chapter 12: Numerical Integration
   Newtown-Cotes Formulas
      Trapezoidal Rule
         Implementation
         Testing the Trapezoidal Method
      Simpson's Rule
         Implementation
         Testing the Simpson Method
      Higher Order Rules
   Romberg Integration
      Implementation
      Testing the Romberg Method
   Gaussian Integration
      Gauss-Legendre Integration
         Implementation
         Testing Gauss-Legendre Integration
      Gauss-Laguerre Integration
         Implementation
         Testing Gauss-Laguerre Integration
      Gauss-Hermite Integration
         Implementation
         Testing Gauss-hermite Integration
      Gauss-Chebyshev Integration
         Implementation
         Testing Gauss-Chebyshev Integration

Chapter 13: Ordinary Differential Equations
   Euler Method
      Implementation
      testing Euler's Method
   Second-Order Runge-Kutta Method
      Implementation
      Testing the Second-Order Runge-Kutta Method
   Fourth-Order Runge-Kutta Method
      Implementation
      Testing the Fourth-Order Runge-Kutta Method
   Adaptive Runge-Kutta Method
      Implementation
      Testing the Adaptive Runge-Kutta Method
   Runge-Kutta Method for Systems
      Implementation
      Testing the MultiRungeKutta4 Method
   Adaptive Runge-Kutta Method for Systems
      Implementation
      Testing the MultiRungeKuttaFehlberg Method

Chapter 14: Boundary Value Problems
   Shooting Method
      Implementation
      Testing the Shooting2 Method
   Finite Difference for Linear Equation
      Algorithm
      Implementation
      Testing FiniteDifferenceLinear2 Method
   Finite Difference for Nonlinear Equation
      Algorithm
      Implementation
      Testing FiniteDifferenceNonlinear2 Method
   Finite Difference for Higher-Order Equation
      Algorithm
      Implementation
      Testing FiniteDifferenceLinear4 Method

Chapter 15: Eigenvalue Problems
   Jacobi Method
      Algorithm
      Implementation
      Testing the Jacobi Method
   Power Interation
      Algorithm
      Implementation
      Testing the Power Method
   Inverse Interation
      Algorithm
      Implementation
      Testing the Inverse Method
   Rayleigh Method
      Algorithm
      Implementation
      Testing the Rayleigh Method
   Rayleigh-Quotient Method
      Algorithm
      Implementation
      Testing the Rayleigh-Quotient Method
   Matrix Tridiagonalization
      Algorithm
      Implementation
      Testing the Tridiagonalize Method
   Eigenvalues of Symmetric Tridiagonal Matrices
      LU Decomposition of Tridiagonal Matrices
      Implementation
      Examples

Index

Introduction

        Overview
        What This Book Includes
        Is This Book for You
        What Do You Need to Use This Book
        How This Book is Organized
        Using Code Examples

Overview

Welcome to Practical Numerical Methods with C#. This book is intended for scientists, engineers, and .NET developers who want to create scientific and engineering applications using C# and the .NET Framework. For many years, FORTRAN has been the dominant language of scientific and engineering computation. But as Microsoft C# and .NET Framework gain popularity, you may find that they are also suitable for technical computing. This book presents C#-based procedures that perform fundamental mathematical and numerical computations critical to scientists and engineers.

The power of the C# programming language, combined with the simplicity of implementing Windows Forms, WPF desktop applications, and Silverlight Web applications based on Visual Studio .NET framework, makes real-world .NET program development faster and easier than ever before. Visual C# is a versatile and flexible tool that allows users with even the most elementary programming abilities to not only perform complicated computations, but also to display the calculated results in a variety of graphical representations. In this regard, C# is more powerful than FORTRAN, because it is hard to show results graphically using FORTRAN.

The main advantage of using FORTRAN in scientific and engineering computing is its rich math libraries. These libraries implement a complete collection of mathematical, statistical, and numerical algorithms, which have been evolving steadily for several decades. Each subroutine and algorithm in these libraries has undergone rigorous testing and quality assurance, providing users with more time to focus on their applications. On the other hand, the C# programming language is relatively new to the scientific and engineering community. The lack of C# math libraries prevents many researchers from using the C# programming language in their applications. In this book, I will show you that it is fairly easy to develop math libraries in C#, and that it is worth developing scientific and engineering applications using C# due to its computing power and graphical representations capability.

The aim of this book is to provide scientists and engineers with a comprehensive explanation of scientific computing using C#. Much of the work in this book is original, based on my own programming experience in developing commercial Computer Aided Design (CAD) packages, which involve intensive scientific computations and sophisticated graphical representations. With FORTRAN, developing advanced graphics and chart applications is a difficult and time-consuming task. To add even simple charts or graphs to your applications, you have to waste either effort creating a chart program, or money buying commercial graphics and chart add-on packages. Visual C# and its rich graphics features make it possible to easily implement both powerful math libraries and professional graphics using entirely managed C# codes.

Practical Numerical Methods with C# provides an in-depth introduction to performing complicated scientific computations using C# applications. In this book, I will begin with an overview of the C# and .NET Framework, and then present procedural descriptions of linear algebra, numerical solution of nonlinear and ordinary differential equations, optimization, parameter estimation, and special functions of mathematical physics. I will show you how to create useful C# mathematical and numerical libraries that you can use in real-world scientific and engineering problems. I will try my best to introduce the C# program to scientists and engineers in a simple way--simple enough for C# beginners to follow with ease. From this book, you will learn how to perform complicated scientific computations and create your own math libraries based on C# and the .NET Framework.

Practical Numerical Methods with C# is not simply a book, but a powerful C# math library. You will find that you can immediately use some of the examples in this book to solve your real-world problems, and that you can use others to give you inspiration for adding more advanced math libraries to your applications.

What This Book Includes

This book and its sample code listings, which are available for download at this website, provide you with:

• Complete, in-depth instruction on practical scientific computing and programming using C#. After reading this book and running the example programs contained within, you will be able to create various math and numerical libraries in your own C# applications.
• Ready-to-run example programs that allow scientists and engineers to explore the numerical methods described in the book. You can use these examples to better understand how the mathematical models and algorithms work. You can also modify the code or add new features to them to form the basis of your own programs. Some of the example code listings provided in this book are already sophisticated math libraries; these can be used directly in your own real-world applications.
• Many C# classes in the sample code listings that you will find useful in solving your real-world scientific and engineering problems. These classes include linear algebra, matrix manipulation, numerical approaches, and other useful utility classes. You can extract these classes and plug them into your own applications.

Is This Book for You

You do not have to be an experienced C# developer or expert to use this book. I designed this book to be useful to scientists and engineers of all levels of C# programming experience. In fact, I believe that if you have some experience with programming languages other than C#, you will be able to sit down in front of your computer, start up the Microsoft .NET Framework SDK or Visual Studio .NET, follow the examples provided with this book, and quickly become familiar with C# programming in scientific computing. For those of you who are already experienced C# developers, I believe this book has plenty to offer to you as well. The information in this book about creating C# math libraries is not available in any other C# tutorial and reference book. In addition, most of the example programs provided in this book can be used directly in your own real-world application development. This book will provide you with a level of detail, explanation, instruction, and sample program code that will enable you to do just about anything scientific and engineering computing-related with Visual C#.

This book is specifically designed for scientists and engineers. In fact, my own background is in theoretical physics, a field involving extensive numerical calculations as well as graphical representations of calculated data. I have been dedicated to this field for many years. My first computer experience was with FORTRAN. Later on, I gained programming experience in Basic, C, C++, and MATLAB. I still remember how hard it was in those early days to present computational results graphically. I often spent hours creating a publication-quality chart by hand, using a ruler, graph paper, and rub-off lettering. During that time, I started to pay attention to various development tools that could be used to create integrated applications. I tried to find an ideal development tool that would allow me to not only easily generate data (computation capability) but also easily represent data graphically (graphics and chart power). The C# and Microsoft Visual Studio .NET development environment makes it possible to develop such integrated applications. Ever since Microsoft .NET 1.0 came out, I have been in love with the C# language, and have used this tool to successfully create powerful scientific and plotting applications, including commercial CAD packages.

The majority of the example programs in this book can be used routinely by scientists and engineers. Throughout this book, I will emphasize the usefulness of C# programming in solving real-world scientific and engineering problems. If you follow this book closely, you will be able to easily develop various math and numerical libraries. At the same time, I will not spend too much time discussing program style, execution speed, and code optimization, because there is already a plethora of books out there that deal with these topics. Most of the example programs in this book omit error handling. This makes the code easier to understand by focusing on the key concepts.

Note that this book focuses on numerical computing methods and math library development using C#. It will not address the graphical representations of calculation results, even though the real power of the .NET Framework is its ability to create graphics and user interfaces. If you are interested in graphics and user interface programming in C#, you can read my other books:

Practical C# Charts and Graphics - This book is a perfect guide to learning all the basics for creating advanced chart and graphics applications in C#, GDI+, and Windows Form. It clearly explains practical chart and graphics methods and their underlying algorithms. The 2D and 3D chart packages contained in the book can be directly used in your C# applications.

Practical WPF Graphics Programming - This book provides all the information you need to add advanced graphics to your .NET applications using C# and Windows Presentation Foundation (WPF), which comes with the new version (3.0 or later) of .NET framework. From 2D shapes and charts to complex interactive 3D models, this book uses code examples to explain every step it takes to build a variety of WPF graphics applications.

Practical Silverlight Programming - This book shows you how to develop rich interactive applications (RIAs) for Web using C# and Silverlight. Silverlight is a subset of WPF and enables you to create advanced graphics and user interfaces for Web applications. You will learn from this book how to display your computation results graphically and interactively over the internet.

What Do You Need to Use This Book

You will need no special equipment to make the best use of this book and understand the algorithms. To run and modify the sample programs, you will need a computer capable of running either the Windows Vista or XP operating system. The software installed on your computer should include .NET Framwork SDK 2.0 or later, which is available for free at the Microsoft Website. It is best to have Visual Studio 2005 or 2008 standard edition or higher. Please note that all of the example programs and math libraries were created and tested in Visual Studio 2008. However, the example code should be independent of the platform you use.

How This Book is Organized

This book is organized into fifteen chapters, each of which covers a different topic of numerical computing. The following summaries of each chapter should give you an overview of the book?s contents:

Chapter 1, Overview of C# Programming
This chapter introduces the basics of C# programming, including the basic types, properties, methods, mathematical operations, and how to create branches and loops.

Chapter 2, Complex Numbers and Functions
This chapter demonstrates how to implement a complex structure, which contains the definition of complex numbers, complex operators, and commonly used complex functions. This structure allows you to perform various computations using complex numbers and functions.

Chapter 3, Vectors and Matrices
This chapter introduces a more general n-dimensional vector class and a general matrix class with the n×m dimension, which can be used in many scientific and engineering computations involving the solution of linear equations with multiple variables. Matrix analysis is a basic theory of these linear operations.

Chapter 4, LinearAlgebraic Equations
This chapter introduces various numerical methods for solving linear equations with an arbitrary number of unknowns. Solving linear equations is one of the most commonly used operations in numerical analysis and scientific and engineering applications.

Chapter 5, Nonlinear Equations
This chapter describes several numerical methods for solving nonlinear equations. These numerical methods are all iterative in nature, and may be used for equations that contain one or several variables.

Chapter 6, Special Functions
This chapter discusses a special function class, which contains popular special functions such as the gamma function, beta function, error function, elliptic integral, Laguerre function, Hermit function, Chebyshev function, Legendre function, and Bessel function, among others.

Chapter 7, Random Numbers and Distribution Functions
This chapter covers a variety of random number generators and different probability distribution functions, which can be used to simulate the different chaotic circumstances that can be found in the real world.

Chapter 8, Interpolation
This chapter explains the implementation of several interpolation methods, which can be used to construct new data points within the range of a discrete set of known data points. Interpolation can be regarded as a special case of curve fitting, in which the function must go exactly through the data points.

Chapter 9, Curve Fitting
This chapter explains a variety of curve fitting approaches that can be applied to data containing noise, usually due to measurement errors. Curve fitting tries to find the best fit to a set of given data. Thus, the curve does not necessarily pass through all of the given data points.

Chapter 10, Optimization
This chapter covers several popular methods for optimizing functions with multiple variables, including the golden search, Newton, simplex, simulated annealing, and differential evolution techniques. In particular, simulated annealing and differential evolution can deal with highly nonlinear models, chaotic and noisy data, and constraints.

Chapter 11, Numerical Differentiation
This chapter discusses several methods of numerical differentiation, such as forward and backward difference, central difference, extended central difference, Richardson extrapolation, and derivatives by interpolation. These methods provide you with different tools for estimating the derivative of a function.

Chapter 12, Numerical Integration
This chapter covers a variety of methods for numerical integration, including methods based on Newton-Cotes formulas, Romberg integration, and Gaussian quadrature methods. These methods can be used to estimate the finite and infinite integrals of functions.

Chapter 13, Ordinary Differential Equations
This chapter focuses on solving ordinary differential equations numerically. It presents several popular methods including the Euler method, second- and fourth-order Runge-Kutta methods, the adaptive Runge-Kutta method, and the Runge-Kutta methods that can be used to solve a system of ordinary differential equations.

Chapter 14, Boundary Value Problems
This chapter discusses two methods for solving boundary value problems: the shooting method and the finite differences method. The shooting method involves guessing the missing values, and the resulting solution is very unlikely to satisfy boundary conditions at the other end. The finite difference method involves approximating the differential equations by finite differences at evenly spaced mesh points.

Chapter 15, Eigenvalue Problems
This chapter presents several popular methods for solving eigenvalue problems, including the Jacobi method, power iteration, Rayleigh method, Rayleigh-quotient method, and matrix tridiagonalization method. These methods offer you nontrivial tools for calculating eigenvalues and eigenvectors of real symmetric matrix systems.

Using Code Examples

You may use the code in this book in your applications and documentation. You do not need to contact me or the publisher for permission unless you are reproducing a significant portion of the code. For example, writing a program that uses several chunks of code from this book does not require permission. Selling or distributing the example code listings does require permission. Incorporating a significant amount of example code from this book into your applications and documentation does require permission. Integrating the example code from this book into your commercial products is not allowed without the written permission from the author.