C#--解方程组之Seidel迭代法

简介

高斯-赛德尔迭代法是解 线性方程组的常用迭代法之一，设线性方程组为

高斯-赛德尔迭代法的迭代公式为

当然，此处假定

，在很多情况下，它比简单迭代法收敛快，它和简单迭代法的不同点在于计算

时，利用了刚刚迭代出的

的值，当系数矩阵 A 严格对角占优或对称正定时，高斯-赛德尔迭代法必收敛。

迭代公式的实现

 public void Calcu6()
        {
            int count1 = 0, count2 = 0;
            while (true)
            {
                for (int i = 0; i < n; i++)
                {
                    double sum1 = 0,sum2=0;
                    for (int j = i+1; j <n; j++)
                    {
                        sum1 += a[i, j] * x[j];
                    }
                    for(int j=0;j<i-1;j++)
                    {
                        sum2 += a[i, j] * x2[j];
                    }
                    x[i] = (a[i, n] - sum2-sum1) / a[i, i];
                    if (Math.Abs(x2[i] - x[i]) < e)
                        count2++;
                }
                count1++;
                if (count1 > 10000)
                { Console.WriteLine("迭代发散！！！"); break; }
                if (count2 == n)
                { Console.WriteLine("迭代次数：{0}", count2); break; }
            }
        }

源程序：

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace Gauss_Seidel迭代
{
    class Seidel
    {
        int n;
        public int N
        {
            get { return n; }
            set { n = value; }
        }
        double[,] a;
        public double[,] A
        {
            get { return a; }
            set { a = value; }
        }
        double[] x;
        public double[] X
        {
            get { return x; }
            set { x = value; }
        }
        double e = 0.00001;
        public double E
        {
            get { return e; }
            set { e = value; }
        }
        private double[] x2;
        public double[] X2
        {
            get { return x2; }
            set { x2 = value; }
        }
        public void Input()
        {
            Console.WriteLine("请输入阶数：");
            n = Convert.ToInt32(Console.ReadLine());
            a = new double[n, n + 1];
            x = new double[n];
            x2 = new double[N + 1];
            for (int i = 0; i < N; i++)
            {
                x2[i]=x[i];
            }
            Console.WriteLine("请输入各行系数(','或' '隔开)：");
            for (int i = 0; i < n; i++)
            {
                string s = Console.ReadLine();
                string[] ss = s.Split(' ', ',');
                for (int j = 0; j < n + 1; j++)
                {
                    a[i, j] = Convert.ToDouble(ss[j]);
                }
            }
        }
        public void Calcu6()
        {
            int count1 = 0, count2 = 0;
            while (true)
            {
                for (int i = 0; i < n; i++)
                {
                    double sum1 = 0,sum2=0;
                    for (int j = i+1; j <n; j++)
                    {
                        sum1 += a[i, j] * x[j];
                    }
                    for(int j=0;j<i-1;j++)
                    {
                        sum2 += a[i, j] * x2[j];
                    }
                    x[i] = (a[i, n] - sum2-sum1) / a[i, i];
                    if (Math.Abs(x2[i] - x[i]) < e)
                        count2++;
                }
                count1++;
                if (count1 > 10000)
                { Console.WriteLine("迭代发散！！！"); break; }
                if (count2 == n)
                { Console.WriteLine("迭代次数：{0}", count2); break; }
            }
        }
        public void Output()
        {
            Console.WriteLine("方程系数为：");
            for (int i = 0; i < n; i++)
            {
                string s = null;
                for (int j = 0; j < n + 1; j++)
                {
                    s += string.Format("{0,8:f2}", a[i, j]);
                }
                Console.WriteLine(s);
            }
        }

        public void OutputX()
        {
            Console.WriteLine("\n方程组的解是：");
            for (int i = 0; i < n; i++)
            {
                Console.WriteLine("x{0}={1}", i + 1, x[i]);
            }
        }
    }
    class Program
    {
        static void Main(string[] args)
        {
            Seidel abc = new Seidel();
            abc.Input();
            abc.Output();
            abc.Calcu6();
            abc.OutputX();
        }
    }
}

运行结果：

谢谢！！！