C#是一种面向过程的算法设计语言,在处理数学计算时显得力不从心。面向集合是一种比较高阶的特性,允许程序直接对集合进行直接处理,而不用显示进行循环调用,不需要指定路径,因此可以大幅提高开发效率,进行并行优化。很多数学软件,如MatLab是面向矩阵的,而开源语言R是面向向量的,SQL是面向关系系的、APL(Array processing language)是一种一种多用途、第三代(3GL)编程语言,在向量、矩阵等各种秩的数组处理上非常简单。SPSS,SAS等都需要大量的集合运算。
本文试图从C#本身的特性出发,用C#模拟面向集合的方法。
问题引出
C#是一种面向过程的算法设计语言,在处理数学计算时显得力不从心。面向集合是一种比较高阶的特性,允许程序直接对集合进行直接处理,而不用显示进行循环调用,不需要指定路径,因此可以大幅提高开发效率,进行并行优化。很多数学软件,如MatLab是面向矩阵的,而开源语言R是面向向量的,SQL是面向关系系的、APL(Array processing language)是一种一种多用途、第三代(3GL)编程语言,在向量、矩阵等各种秩的数组处理上非常简单。SPSS,SAS等都需要大量的集合运算。
本文试图从C#本身的特性出发,用C#模拟面向集合的方法。
更期望C#能向MatLab, APL,R那样直接处理集合运算,进入科学和工程计算领域,为以后的并行计算奠定基础。
有一列观测值,用List存储,我们现在需要求出每一个观测值的正弦Sin值。
用C#面向过程的语法表示如下:
List<double> list2;
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for (int i = 0; i < list2.Count; i++)
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list2[i] = Math.Sin(list2[i]);
求Sin值,是一个繁琐而又重复的问题。我们希望Math.Sin(Collection c),在不改变已有代码(不扩展Math.Sin)的情况下,自动处理集合,就像在MatLab里面。
C#是面向过程的,而Matlab是面向矩阵的,SQL是面向关系代数的。关系代数和矩阵,都可以看作集合的特例。(LINQ部分加入了面向集合的特性)
面向过程,需要程序员书写算法的每一个过程和细节,指明执行路径,这主要表现在循环语句的使用上(包括for, foreach, while…)。
面向过程给了程序员最充分的自由和最大的灵活,但其固有的“底层”,导致了开发效率的底下,同时不利于并行计算和系统优化。而在数学中,大部分计算都是基于矩阵(集合),例如图形图像处理,概率论,数理统计,优化控制等等。 所以C#难以胜任运算集中和知识处理,人工智能设计。
由于C#实在是太优美,是目前最艺术的语言,利用C#现有特性,我们可以简单的模拟前面提出的问题
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Code
public static List<double> Apply(Converter<double, double> f, List<double> l)
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{
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List<double> list2 = new List<double>(l);
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for (int i = 0; i < list2.Count; i++)
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list2[i] = Math.Sin(list2[i]);
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for (int i = 0; i < l.Count; i++)
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{
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list2[i] = f(l[i]);
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}
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return list2;
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}
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这样,我们可以在Apply来处理一些关于集合处理的问题。
下面在给出一个处理矩阵的例子:
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Code
public static Matrix Apply(Converter<double, double> f, Matrix m)
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{
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Matrix m2 = new Matrix(m);
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for (int i = 0; i < m.Row; i++)
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for (int j = 0; j < m.Col; j++)
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m2[i, j] = f(m2[i, j]);
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return m2;
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}
使用这个Apply,可以处理矩阵集合相关的计算。
矩阵定义如下:
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Code
public class Matrix
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{
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public double[,] data;
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public Matrix(int row, int col)
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{
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data = new double[row, col];
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}
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//复制构造函数
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public Matrix(Matrix m)
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{
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data = (double[,])m.data.Clone();
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}
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public int Col
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{
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get
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{
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return data.GetLength(1);
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}
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}
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// 行数
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public int Row
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{
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get
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{
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return data.GetLength(0);
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}
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}
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//重载索引
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//存取数据成员
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public virtual double this[int row, int col]
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{
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get
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{
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return data[row, col];
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}
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set
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{
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data[row, col] = value;
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}
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}
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//维数
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public int Dimension
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{
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get
{ return 2; }
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}
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public string ToString(int prec)
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{
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StringBuilder sb = new StringBuilder();
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string format = "{0:F" + prec.ToString() + "} ";
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for (int i = 0; i < Row; i++)
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{
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for (int j = 0; j < Col - 1; j++)
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{
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sb.Append(string.Format(format, data[i, j]));
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}
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sb.Append(data[i, Col - 1]);
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sb.Append(""n");
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}
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return sb.ToString();
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}
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}
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再看下面复数的例子:
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Code
public static List<double> Apply(Converter< Complex,double> f, List<Complex> l)
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{
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List<double> l2 = new List<double>(l.Count);
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for (int i = 0; i < l.Count; i++)
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l2.Add(f(l[i]));
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return l2;
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}
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使用这个Apply,可以处理复数集合相关的许多计算。
复数类的定义如下:
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Code
public class Complex:ICloneable
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{
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private double real;
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/**//// <summary>
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/// 复数的实部
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/// </summary>
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public double Real
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{
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get
{ return real; }
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set
{ real = value; }
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}
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private double image;
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/**//// <summary>
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/// 复数的虚部
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/// </summary>
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public double Image
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{
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get
{ return image; }
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set
{ image = value; }
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}
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/**//// <summary>
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/// 默认构造函数
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/// </summary>
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public Complex()
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: this(0, 0)
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{
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}
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/**//// <summary>
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/// 只有实部的构造函数
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/// </summary>
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/// <param name="real">实部</param>
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public Complex(double real)
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: this(real, 0)
{ }
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/**//// <summary>
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/// 由实部和虚部构造
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/// </summary>
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/// <param name="real">实部</param>
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/// <param name="image">虚部</param>
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public Complex(double r, double i)
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{
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real = r;
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image = i;
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}
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/**////重载加法
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public static Complex operator +(Complex c1, Complex c2)
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{
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return new Complex(c1.real + c2.real, c1.image + c2.image);
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}
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/**////重载减法
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public static Complex operator -(Complex c1, Complex c2)
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{
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return new Complex(c1.real - c2.real, c1.image - c2.image);
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}
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/**////重载乘法
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public static Complex operator *(Complex c1, Complex c2)
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{
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return new Complex(c1.real * c2.real - c1.image * c2.image, c1.image * c2.real + c1.real * c2.image);
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}
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/**//// <summary>
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/// 求复数的模
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/// </summary>
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/// <returns>模</returns>
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public double Modul
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{
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get
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{
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return Math.Sqrt(real * real + image * image);
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}
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}
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public static double Sin(Complex c)
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{
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return c.image / c.Modul;
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}
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/**//// <summary>
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/// 重载ToString方法
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/// </summary>
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/// <returns>打印字符串</returns>
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public override string ToString()
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{
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if (Real == 0 && Image == 0)
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{
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return string.Format("{0}", 0);
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}
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if (Real == 0 && (Image != 1 && Image != -1))
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{
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return string.Format("{0} i", Image);
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}
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if (Image == 0)
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{
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return string.Format("{0}", Real);
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}
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if (Image == 1)
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{
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return string.Format("i");
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}
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if (Image == -1)
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{
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return string.Format("- i");
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}
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if (Image < 0)
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{
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return string.Format("{0} - {1} i", Real, -Image);
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}
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return string.Format("{0} + {1} i", Real, Image);
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}
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ICloneable 成员#region ICloneable 成员
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public object Clone()
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{
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Complex c = new Complex(real, image);
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return c;
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}
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#endregion
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}
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从前面三个例子,我们可以看出,集合有多种表示方式,有.net框架中的List,也有自定义的Matrix,同时集合的元素也是多种数据类型,有系统中的值类型,也有自定义的复数Complex类型。
当然这种算法过于勉强,显然不是我们所需要的。
我们需要的是一个在不更改现有语言的情况下,不扩充Math.Sin函数(试着想想有多少个类似的函数,Cos, Tan, 我们自己定义的各种函数)。系统自动处理集合。也就是说,对于函数 public delegate TOutput Converter<TInput, TOutput>(TInput input);public T1 Func<T1, T2>(T2 e); Func是Converter的实例。只要Func能够处理原子类型,那么就能处理自动所有的原子类型构成的任意集合,而不需要程序员去写多余的代码。
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