高斯列主元素消去法解多元一次方程组

  1 /********************************
  2 高斯列主元素消去法解多元一次方程组
  3  * 
  4  * 例如 方程组
  5  * x+y+z=7
  6  * 2x+y-z=6
  7  * x-y-2z=4
  8  * 
  9  * 矩阵阶数N=3
 10  * a[N,N]为1  1  1
 11  *         2  1 -1
 12  *         1 -1 -2
 13  * b[N]为  7
 14  *         6
 15  *         4
 16  *********************************/
 17 using System;
 18 using System.Collections.Generic;
 19 using System.Linq;
 20 using System.Text;
 21 using System.Threading.Tasks;
 22 
 23 namespace Gauss
 24 {
 25     class Program
 26     {
 27         const int MaxN = 20;
 28         private float[,] a = new float[MaxN, MaxN];
 29         private float[] b = new float[MaxN];
 30         private float[] x = new float[MaxN];
 31 
 32         private int N;//所要计算的矩阵的阶数
 33         private float center;//中间变量
 34         static void Main(string[] args)
 35         {
 36             Program p = new Program();
 37             p.ReadMatrix();
 38             p.ShowMatrix();
 39             p.Calculate();
 40             p.ShowResult();
 41             Console.ReadLine();
 42         }
 43 
 44         //输入线性方程组对应的矩阵
 45         private void ReadMatrix()
 46         {
 47             Console.Write("请输入矩阵的阶数：");
 48             while (!int.TryParse(Console.ReadLine(),out N))
 49             {
 50                 Console.WriteLine("请输入正确的阶数！");
 51             }
 52 
 53             Console.WriteLine("请输入矩阵a[{0}][{0}]：",N);
 54             for (int i = 0; i < N; i++)
 55             {
 56                 for (int j = 0; j < N; j++)
 57                 {
 58                     while (!float.TryParse(Console.ReadLine(), out a[i,j]))
 59                         Console.WriteLine("请正确输入！");
 60 
 61                 }
 62             }
 63 
 64             for (int i = 0; i < N; i++)
 65             {
 66                 Console.Write("请输入矩阵b[{0}]：", i);
 67                 while (!float.TryParse(Console.ReadLine(), out b[i]))
 68                     Console.WriteLine("请正确输入！");
 69             }
 70             Console.WriteLine("\n==========================================");
 71         }
 72 
 73         //显示矩阵对应的方程组
 74         private void ShowMatrix()
 75         {
 76             Console.WriteLine("要求的方程组为：");
 77             for (int i = 0; i < N; i++)
 78             {
 79                 for (int j = 0; j < N; j++)
 80                 {
 81                     if (j == 0) 
 82                     {
 83                         string s;
 84                         s = a[i, j] + "x" + (j + 1);
 85                         Console.Write(s);
 86                     }
 87                         
 88                     else 
 89                     {
 90                         string s = a[i, j] > 0 ? "+" + a[i, j] + "x" + (j+1) : a[i, j] + "x" + (j+1);
 91                         Console.Write(s);
 92                     }
 93                 }
 94                 Console.WriteLine(" = "+b[i]);
 95             }
 96             Console.WriteLine("\n==========================================");
 97         }
 98 
 99         private void Calculate()
100         {
101             float sum;
102             int i, k, p;
103             for (k = 0; k < N-1; k++)
104             {
105                 center = Math.Abs(a[k,k]);//对角线上
106                 for (i = k + 1, p = k; i < N; i++)
107                 {
108                     //选取主元
109                     if (center < Math.Abs(a[i, k]))
110                     {
111                         center = Math.Abs(a[i, k]);
112                         p = i;
113                     }
114                 }
115 
116                 if (i > k)
117                 {
118                     //移动主元
119                     for (int m = k; m < N; m++)
120                     {
121                         center = a[k, m];
122                         a[k, m] = a[p, m];
123                         a[p, m] = center;
124                     }
125                     center = b[k];
126                     b[k] = b[p];
127                     b[p] = center;
128                 }
129 
130                 //消元
131                 for (int m = k + 1; m < N; m++)
132                 {
133                     center = a[m, k];
134                     for (int n = k; n < N; n++)
135                     {
136                         a[m, n] -= center * a[k, n] / a[k, k];
137                     }
138                     b[m] -= center * b[k] / a[k, k];
139                 }    
140 
141                     
142             }
143             //x[N]的值
144             x[N - 1] = b[N - 1] / a[N - 1, N - 1];
145             for (int m = N - 2; m >= 0; m--)
146             {
147                 sum = b[m];
148                 for (int j = N - 1; j > m; j--)
149                 {
150                     sum -= a[m, j] * x[j];
151                     x[m] = sum / a[m, m];
152                 }
153             }
154         }
155 
156 
157         private void ShowResult()
158         {
159             Console.WriteLine("结果为：");
160             for (int i = 0; i < N; i++)
161             {
162                 Console.Write("x{0}={1}\n",i+1,x[i]);
163             }
164             
165         }
166     }
167 }