数值分析实验四 线性方程组数值解法

一、实验目的

1、熟悉求解线性方程组的有关理论和方法；
2、会编制LU 分解法、雅可比及高斯—塞德尔迭代法德程序；
3、通过实际计算，进一步了解各种方法的优缺点，选择合适的数值方法。

二、算法描述

3.1 矩阵直接三角分解法
算法
将方程组Ax=b 中的A分解为A=LU，其中L为单位下三角矩阵，U为上三角矩阵，则方程组Ax=b化为解2个方程组Ly=b，Ux=y，具体算法如下：



3.2 迭代法
3.2.1 雅可比迭代法
算法:设方程组Ax=b系数矩阵的对角线元素,M为迭代次数容许的最大值,ε为容许误差。

3.2.2 高斯-塞尔德迭代法
算法:设方程组Ax=b的系数矩阵的对角线元素,,M为迭代次数容许的最大值,ε为容许误差

三、源程序

1、矩阵直接三角分解法

#include 
#include 
#include 
using namespace std;

/*
1 2 -12 8
5 4 7 -2
-3 7 9 5
6 -12 -8 3

27 4 11 49
*/
int main()
{
	cout << "矩阵直接三角分解法\n\n请输入矩阵阶数：" << endl;
	int n;
	cin >> n;
	//数组a[n+1][n+1];
	int** a = new int* [n+1];
	for (int i = 0; i <= n; i++)
	{
		a[i] = new int[n+1];
	}
	//上三角矩阵
	double** u = new double* [n + 1];
	for (int i = 0; i <= n; i++)
	{
		u[i] = new double[n + 1];
	}
	for (int i = 1; i <= n; i++)
	{
		for (int j = 1; j <= n; j++)
		{
			u[i][j]=0;
		}
	}
	//单位下三角矩阵
	double** l = new double* [n + 1];
	for (int i = 0; i <= n; i++)
	{
		l[i] = new double[n + 1];
	}
	for (int i = 1; i <= n; i++)
	{
		for (int j = 1; j <= n; j++)
		{
			if(i==j)l[i][j] = 1;
			else l[i][j] = 0;
		}
	}

	double* b = new double[n+1];
	double* y = new double[n + 1];
	double* x = new double[n + 1];
	
	
	cout << "请输入矩阵A的值：" << endl;
	for (int i = 1; i<=n; i++)
	{
		for (int j =1 ; j<=n; j++)
		{
			cin >> a[i][j];
		}
	}
	cout << "请输入矩阵b的值：" << endl;
	for (int i = 1; i<=n; i++)
	{
		cin >> b[i];
	}

	for (int j = 1; j <= n; j++)
	{
		u[1][j] = a[1][j];
	}
	for (int i = 2; i <= n; i++)
	{
		l[i][1] = a[i][1]*1.0 / a[1][1];
	}
	for (int k = 1; k <= n; k++)
	{
		for (int j = k; j <= n; j++)
		{
			u[k][j] = a[k][j];
			for (int q = 1; q <= k - 1; q++)
			{
				u[k][j] -= l[k][q] * u[q][j];
			}
		}
		for (int i = k + 1; i <= n; i++)
		{
			l[i][k] = a[i][k];
			for (int q = 1; q <= k - 1; q++)
			{
				l[i][k] -= l[i][q] * u[q][k];
			}
			l[i][k] = l[i][k]*1.0/u[k][k];
		}
	}
	
	y[1] = b[1];
	for (int k = 2; k <= n; k++)
	{
		y[k] = b[k];
		for (int q = 1; q <= k - 1; q++)
		{
			y[k] -= l[k][q] * y[q];
		}
	}
	x[n] = y[n]*1.0 / u[n][n];
	for (int k = n - 1; k >= 1; k--)
	{
		x[k] = y[k];
		for (int q = k + 1; q <= n; q++)
		{
			x[k] -= u[k][q] * x[q];
		}
		x[k] = x[k] *1.0/u[k][k];
	}
	cout << "结果为：" << endl;
	for (int i = 1; i <= n; i++)
	{
		cout << "X[" << i-1 << "]=" << setiosflags(ios::fixed) << setprecision(6) << x[i]<<endl;
	}
	return 0;
}

2、雅可比迭代法

#include 
#include 
#include 
using namespace std;

/*
5 2 1
2 8 -3
1 -3 -6

8 21 1
*/
int main()
{
	cout << "雅可比迭代法\n\n请输入矩阵阶数：" << endl;
	int n,count,e;
	cin >> n;
	//数组a[n+1][n+1];
	int** a = new int* [n + 1];
	for (int i = 0; i <= n; i++)
	{
		a[i] = new int[n + 1];
	}
	
	double* b = new double[n + 1];
	cout << "请输入矩阵A的值：" << endl;
	for (int i = 1; i <= n; i++)
	{
		for (int j = 1; j <= n; j++)
		{
			cin >> a[i][j];
		}
	}
	for (int i = 1; i <= n; i++)
	{
		for (int j = 1; j <= n; j++)
		{
			if (i == j && a[i][j] == 0)
			{
				cout << "对角线元素不能为0，请重新输入！" << endl;
				return 0;
			}
		}
	}
	
	cout << "请输入矩阵b的值：" << endl;
	for (int i = 1; i <= n; i++)
	{
		cin >> b[i];
	}
	cout << "请输入迭代次数允许的最大值：" << endl;
	cin >> count;
	cout << "请输入容许误差：" << endl;
	cin >> e;

	double** x = new double* [n + 1];
	for (int i = 0; i <= n; i++)
	{
		x[i] = new double[count+1];
	}
	double* x1 = new double[n + 1];//最终结果
	//设置初始向量
	for (int i = 1; i <= n; i++)
	{
		x[i][0]=0;
	}	
	int k,num=0;
	for (k = 0;;)
	{
		int sum = 0;
		for (int i = 1; i <= n; i++)
		{
			x[i][k + 1] = b[i];
			for (int j = 1; j <= n; j++)
			{
				if (j != i)
				{
					x[i][k + 1] -= a[i][j] * x[j][k];
				}
			}
			x[i][k + 1] /= a[i][i];
			x1[i-1] = x[i][k + 1];
			sum += abs(x[i][k + 1] - x[i][k]);
		}
		if (sum >= e)
		{
			if (k >= count)
			{
				//cout << "超出迭代次数允许的最大值，程序结束";
				break;
			}
			else
			{
				k = k + 1;
			}
		}
	}
	for (int i = 0; i <n; i++)
	{
		cout << "X[" << i << "]=" << setiosflags(ios::fixed) << setprecision(6) << x1[i] << endl;
	}
	return 0;
}

3、高斯-赛德尔迭代法

#include 
#include 
#include 
using namespace std;

/*
8 -3 2
4 11 -1
6 3 12

20 33 36
*/
int main()
{
	cout << "高斯-赛德尔迭代法\n\n请输入矩阵阶数：" << endl;
	int n, count, e;
	cin >> n;
	//数组a[n+1][n+1];
	int** a = new int* [n + 1];
	for (int i = 0; i <= n; i++)
	{
		a[i] = new int[n + 1];
	}

	double* b = new double[n + 1];
	cout << "请输入矩阵A的值：" << endl;
	for (int i = 1; i <= n; i++)
	{
		for (int j = 1; j <= n; j++)
		{
			cin >> a[i][j];
		}
	}
	for (int i = 1; i <= n; i++)
	{
		for (int j = 1; j <= n; j++)
		{
			if (i == j && a[i][j] == 0)
			{
				cout << "对角线元素不能为0，请重新输入！" << endl;
				return 0;
			}
		}
	}

	cout << "请输入矩阵b的值：" << endl;
	for (int i = 1; i <= n; i++)
	{
		cin >> b[i];
	}
	cout << "请输入迭代次数允许的最大值：" << endl;
	cin >> count;
	cout << "请输入容许误差：" << endl;
	cin >> e;

	double** x = new double* [n + 1];
	for (int i = 0; i <= n; i++)
	{
		x[i] = new double[count + 1];
	}
	double* x1 = new double[n + 1];//最终结果
	//设置初始向量
	for (int i = 1; i <= n; i++)
	{
		x[i][0] = 0;
	}
	int k;
	for (k = 0;;)
	{
		int sum = 0;
		for (int i = 1; i <= n; i++)
		{
			x[i][k + 1] = b[i];
			for (int j = 1; j <= i-1; j++)
			{
				x[i][k + 1] -= a[i][j] * x[j][k+1];
			}
			for (int j = i+1; j <= n; j++)
			{
				x[i][k + 1] -= a[i][j] * x[j][k];
			}
			x[i][k + 1] /= a[i][i];
			x1[i - 1] = x[i][k + 1];
			sum += abs(x[i][k + 1] - x[i][k]);
		}
		if (sum >= e)
		{
			if (k >= count)
			{
				//cout << "超出迭代次数允许的最大值，程序结束";
				break;
			}
			else
			{
				k = k + 1;
			}
		}
	}
	for (int i = 0; i < n; i++)
	{
		cout << "X[" << i << "]=" << setiosflags(ios::fixed) << setprecision(6) << x1[i] << endl;
	}
	return 0;
}