C#数值方法与算法工具

数值算法

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"Numerical Methods, Algorithm and Tools in C#" 本书《Numerical Methods, Algorithm and Tools in C#》深入探讨了使用C#编程语言实现数值计算的方法、算法和工具。作者Waldemar Dos Passos提供了丰富的源代码，读者可以直接从CRC Press的官方网站下载，网址为http://www.crcpress.com/product/isbn/9780849374791，选择“Downloads & Updates”选项获取。数值方法在科学计算、工程计算以及各种数据分析领域中扮演着核心角色。C#作为一款功能强大的现代编程语言，具有良好的性能和丰富的库支持，因此成为了实现这些算法的理想选择。书中可能涵盖了以下关键知识点： 1. 基础概念：解释了数值方法的基本原理，包括误差分析、浮点运算精度以及稳定性分析等，这些都是理解和应用数值算法的关键。 2. 线性代数：详述了矩阵运算、特征值问题、线性方程组求解（如高斯消元法、LU分解、QR分解和迭代方法）以及奇异值分解等。 3. 微积分与优化：讨论了数值积分（如梯形法则、辛普森法则）、微分方程求解（如欧拉方法、龙格-库塔方法）以及最优化问题的解决方案（如梯度下降、牛顿法）。 4. 数值微分：介绍了有限差分法和其他近似微分技术，用于求解函数的导数和偏导数。 5. 非线性方程求解：涵盖了迭代方法，如二分法、牛顿法和拟牛顿法，用于找到非线性方程的根。 6. 统计与概率：可能涉及随机数生成、蒙特卡洛模拟和概率分布的数值处理。 7. 特殊函数：介绍如何用C#实现数学中的特殊函数，如贝塞尔函数、伽马函数和傅立叶变换等。 8. 科学计算库：讨论了.NET框架下的科学计算库，如Math.NET Numerics、Accord.NET等，它们为开发者提供了现成的数值算法实现。 9. 高性能计算：可能会涉及并行计算、GPU计算以及如何利用多核处理器提高数值计算的效率。 10. 实际应用：通过实例展示了如何将这些数值方法应用于实际问题，如物理学、工程学、经济学或金融建模等。本书旨在帮助读者掌握C#编程环境下数值计算的核心技术，并能灵活运用到实际项目中。无论你是初学者还是经验丰富的开发者，都能从中受益，提升自己的数值计算能力。通过阅读和实践书中的代码，你将能够更深入地理解数值方法的内在机制，并能有效地解决复杂的计算问题。

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xiv Numerical Methods, Algorithms and Tools in C#

13 Statistical Functions 435

13.1 Introduction . . . . ......................... 435

13.2 SomeUsefulTools ......................... 435

13.3 BasicStatisticalFunctions ..................... 438

13.3.1 MeanandWeightedMean ................. 438

13.3.2 GeometricandWeightedGeometricMean......... 439

13.3.3 HarmonicandWeightedHarmonicMean ......... 440

13.3.4 TruncatedMean....................... 441

13.3.5 RootMeanSquare ..................... 441

13.3.6 Median,RangeandMode ................. 442

13.3.7 MeanDeviation....................... 444

13.3.8 MeanDeviationoftheMean................ 444

13.3.9 MeanDeviationoftheMedian............... 445

13.3.10VarianceandStandardDeviation.............. 445

13.3.11 Moments About the Mean ................. 447

13.3.12Skewness.......................... 448

13.3.13Kurtosis........................... 449

13.3.14CovarianceandCorrelation................. 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 CombinationsandPermutations .................. 464

14.3.1 Combinations........................ 464

14.3.2 Permutations ........................ 467

14.4 GammaFunction .......................... 470

14.5 BetaFunction ............................ 472

14.6 ErrorFunction ............................ 472

14.7 SineandCosineIntegralFunctions ................. 474

14.8 Laguerre Polynomials . . . ..................... 475

14.9 Hermite Polynomials . . . ..................... 476

14.10 Chebyshev Polynomials . . ..................... 477

14.11 Legendre Polynomials . . . ..................... 479

14.12BesselFunctions .......................... 480

15 Curve Fitting Methods 483

15.1 Introduction . . . . ......................... 483

15.2 LeastSquaresFit .......................... 484

15.2.1 Straight-LineFit ...................... 485

15.3 WeightedLeastSquaresFit ..................... 488

15.3.1 WeightedStraight-LineFit................. 488

15.4 LinearRegression .......................... 492

15.4.1 Polynomial Fit . . ..................... 496

Introduction

The main objective of this ﬁrst chapter is to provide my readers with a brief outline

of the .NET Framework, the C# programming language and the basic concepts of

Object Oriented Programming (OOP). Special attention will be given to materials

that illustrate how to best utilize these tools to develop accurate and robust numerical

methods in C# primarily for use in scientiﬁc and engineering applications.

1.1 C# and the .NET Framework

In the late 1990s, Microsoft embarked on a project to update and improve its ﬂag-

ship software application development system, more commonly known as Visual

Studio, and as a result of this effort, an entirely new programming language named

C# emerged that, among other things, essentially incorporates the best features of

all the other popular programming languages of the time such as Java, C/C++ and

Visual Basic. Consequently, since its ﬁrst release in July of 2000, C# has quickly

established itself worldwide as perhaps the preferred language for software applica-

tion development. Besides being a very powerful general purpose, object-oriented

programming language, C# enjoys the full advantage and beneﬁts of being fully in-

tegrated with the Microsoft .NET Framework system.

The Microsoft .NET Framework is a fundamental Windows operating system

component that supports building and running both software applications as well

as Web services. It consists of a large set of class libraries of pre-coded solutions to

common programming problems and also provides a new environment for build-

ing applications that can be deployed and executed across multiple architectures

and operating systems. The .NET Framework was designed to be installed on top

of the Windows operating system and is divided into two key components: a run-

time environment called the Common Language Runtime (CLR), which provides

the runtime services to manage and execute applications originally written in any

one of the .NET programming languages, and a large library of pre-coded object ori-

ented classes called the Framework Class Library (FCL) which provides the required

services for developing .NET applications. Conceptually, .NET applications reside

above the .NET Framework architecture and can be illustrated abstractly as shown

in Table 1.1.

The C# language speciﬁcation [1] has been standardized by ECMA International,

2 Numerical Methods, Algorithms and Tools in C#

TABLE 1.1

Outline of the .NET Framework Architecture

.NET Applications

Visual Basic Visual C# Visual C++

.NET Framework Class Library (FCL)

Common Language Runtime (CLR)

Operating System

which is an industry association dedicated to the standardization of information

and communication systems. As a result, consumers can now choose to buy their

C# compilers from among several different manufacturers such as Microsoft [2],

SharpDevelop [3], DotGNU [4] and Mono [5]. However, the most popular C# com-

piler on the market today comes bundled with Microsoft’s Visual Studio software

development system which, in addition to having a C# compiler, also provides a

full featured integrated development environment (IDE) that standardizes support

for many of the other popular programming languages like Visual Basic and Visual

C++. Accordingly, all the code and examples contained in this book were written

and tested using the latest version of Microsoft Visual Studio.

As part of the natural evolution of programming languages, C# has incorporated

and exploited familiar features from C/C++ and Java that already had a proven record

of success. In addition, C# also has a number of unique new features that make it a

very attractive programming language. For example, C# controls access to hardware

and memory resources and, by default, does not allow for the explicit usage and ma-

nipulation of pointers as C and C++ do except for sections of code that have been

speciﬁcally designated as unsafe. This feature, along with the support of a more pow-

erful garbage collector that automatically manages all aspects of memory allocation

and de-allocation during runtime, has now made frustrating memory leaks and dan-

gling pointers, often hard to ﬁnd and debug in C/C++ programs, a thing of the past.

In addition, improvements made in exception handling provide a well structured

and extensible approach to error detection and recovery. C# is also designed with

type-safe features that make it impossible to have non-initialized variables, to index

arrays beyond their bounds, or to perform unchecked type casts. The C# language

also supports more advanced features such as multi-threading and Just-in-Time (JIT)

compilation from byte code to native code to name a few.

Introduction 3

1.2 Installing C# and the .NET Framework

You can buy Visual C# either by itself or as part of the Visual Studio IDE, which also

includes, in addition to Visual C#, support for other programming languages such as

Visual Basic and Visual C++. Visual C# comes in several editions. If you want

Visual C# all by itself, you have only one choice: the Express edition. However,

if you buy Visual C# as part of the Visual Studio package, you have three choices:

Standard, Professional and Team editions to accommodate every budget, work en-

vironment and skill level. There is also the Academic version of the Professional

edition which is available at a substantial discount for students and teachers. The

key differences between these various editions center primarily on the number of

development environment features available to the programmer. If you do computer

programming as a hobby, the Express edition should work just ﬁne for most of the

applications described in this book. However, if you do a signiﬁcant amount of soft-

ware development in various languages and platforms, then you will likely derive

most beneﬁt from the multipurpose Professional edition.

The Visual C# installation kit may consist of one or more CDs depending on the

edition chosen. The installation itself is relatively easy and is simply a matter of fol-

lowing the directions displayed on the screen. If you do not have the required .NET

Framework already installed on your system, the installation program will perform

that task automatically for you prior to doing anything else. Due to the huge size

of the program, it may take some time to install. But patience is a virtue in pro-

gramming and it begins with the installation of Microsoft Visual Studio. Installing

the latest version of the MSDN reference libraries directly on your computer is also

highly recommended so that help ﬁles can be retrieved and promptly consulted as

needed.

1.3 Overview of Object-Oriented Programming (OOP)

There are primarily two methods of programming in use today: procedural and

object-oriented. Procedural programming has its roots in the earliest forms of pro-

gramming languages and essentially involves creating and naming computer memory

locations that can hold data which can then be changed and manipulated through a

series of sequential steps. The named computer memory locations are called vari-

ables because they hold values that might vary at some point during the life of the

program. Sometimes a ﬁnite number of these sequential steps used in computer

programs can be grouped into smaller logical units called procedures. Hence the

objective of procedural programming is to focus on the creation of procedures to op-

erate and potentially alter data. If, at some later point in time, the program’s original

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