高斯列主元素消去法解多元一次方程组

1、Gauss消去法及列主元素消去法算法介绍

2、c#源码

/********************************
高斯列主元素消去法解多元一次方程组
 * 
 * 例如 方程组
 * x+y+z=7
 * 2x+y-z=6
 * x-y-2z=4
 * 
 * 矩阵阶数N=3
 * a[N,N]为1  1  1
 *         2  1 -1
 *         1 -1 -2
 * b[N]为  7
 *         6
 *         4
 *********************************/
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace Gauss
{
    class Program
    {
        const int MaxN = 20;
        private float[,] a = new float[MaxN, MaxN];
        private float[] b = new float[MaxN];
        private float[] x = new float[MaxN];

        private int N;//所要计算的矩阵的阶数
        private float center;//中间变量
        static void Main(string[] args)
        {
            Program p = new Program();
            p.ReadMatrix();
            p.ShowMatrix();
            p.Calculate();
            p.ShowResult();
            Console.ReadLine();
        }

        //输入线性方程组对应的矩阵
        private void ReadMatrix()
        {
            Console.Write("请输入矩阵的阶数：");
            while (!int.TryParse(Console.ReadLine(),out N))
            {
                Console.WriteLine("请输入正确的阶数！");
            }

            Console.WriteLine("请输入矩阵a[{0}][{0}]：",N);
            for (int i = 0; i < N; i++)
            {
                for (int j = 0; j < N; j++)
                {
                    while (!float.TryParse(Console.ReadLine(), out a[i,j]))
                        Console.WriteLine("请正确输入！");

                }
            }

            for (int i = 0; i < N; i++)
            {
                Console.Write("请输入矩阵b[{0}]：", i);
                while (!float.TryParse(Console.ReadLine(), out b[i]))
                    Console.WriteLine("请正确输入！");
            }
            Console.WriteLine("
==========================================");
        }

        //显示矩阵对应的方程组
        private void ShowMatrix()
        {
            Console.WriteLine("要求的方程组为：");
            for (int i = 0; i < N; i++)
            {
                for (int j = 0; j < N; j++)
                {
                    if (j == 0) 
                    {
                        string s;
                        s = a[i, j] + "x" + (j + 1);
                        Console.Write(s);
                    }
                        
                    else 
                    {
                        string s = a[i, j] > 0 ? "+" + a[i, j] + "x" + (j+1) : a[i, j] + "x" + (j+1);
                        Console.Write(s);
                    }
                }
                Console.WriteLine(" = "+b[i]);
            }
            Console.WriteLine("
==========================================");
        }

        private void Calculate()
        {
            float sum;
            int i, k, p;
            for (k = 0; k < N-1; k++)
            {
                center = Math.Abs(a[k,k]);//对角线上
                for (i = k + 1, p = k; i < N; i++)
                {
                    //选取主元
                    if (center < Math.Abs(a[i, k]))
                    {
                        center = Math.Abs(a[i, k]);
                        p = i;
                    }
                }

                if (i > k)
                {
                    //移动主元
                    for (int m = k; m < N; m++)
                    {
                        center = a[k, m];
                        a[k, m] = a[p, m];
                        a[p, m] = center;
                    }
                    center = b[k];
                    b[k] = b[p];
                    b[p] = center;
                }

                //消元
                for (int m = k + 1; m < N; m++)
                {
                    center = a[m, k];
                    for (int n = k; n < N; n++)
                    {
                        a[m, n] -= center * a[k, n] / a[k, k];
                    }
                    b[m] -= center * b[k] / a[k, k];
                }    

                    
            }
            //x[N]的值
            x[N - 1] = b[N - 1] / a[N - 1, N - 1];
            for (int m = N - 2; m >= 0; m--)
            {
                sum = b[m];
                for (int j = N - 1; j > m; j--)
                {
                    sum -= a[m, j] * x[j];
                    x[m] = sum / a[m, m];
                }
            }
        }


        private void ShowResult()
        {
            Console.WriteLine("结果为：");
            for (int i = 0; i < N; i++)
            {
                Console.Write("x{0}={1}
",i+1,x[i]);
            }
            
        }
    }
}

3、运行结果