列主元素消元法

Posted on | In

列主元素消元法是在高斯消元法方法上的改进,目的是为了减小计算机在消元过程的误差.

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#include <iostream>
#include <iomanip>
#include <cmath>

using namespace std;

//打印矩阵
template<typename T>
void print_mat(T* matrix, int row, int col)
{
	for (int i = 0; i < row; ++i)
	{
		for (int j = 0; j < col; ++j)
			cout << std::left << setw(9) << matrix[i * col + j] << "      ";

		cout << endl;
	}
	cout << endl;
}

//打印向量
template<typename T>
void print_vec(T* vector, int len)
{
	for (int i = 0; i < len; ++i)
		cout << vector[i] << endl;
	cout << endl;
}


//打印增广矩阵
template<typename T, typename I>
void print_matrix_vec(T* matrix, int row, int col, I* vector, int len)
{
	for (int i = 0; i < row; ++i)
	{
		for (int j = 0; j < col; ++j)
			cout << std::left << setw(9) << matrix[i * col + j] << "      ";
		cout << vector[i] << ends;
		cout << endl;
	}
	cout << endl;
}


//交换两行
template<typename T>
void swap_row(T* matrix, int row, int col, int source_row, int dest_row)
{
	//source_row与dest_row交换
	//cout << "第 " << source_row << " 行与第 " << dest_row << " 行互换";
	T temp_val = 0;
	for (int i = 0; i < col; ++i)
	{
		temp_val = matrix[(dest_row - 1) * col + i];
		matrix[(dest_row - 1) * col + i] = matrix[(source_row - 1) * col + i];
		matrix[(source_row - 1) * col + i] = temp_val;
	}
	cout << endl;
}

//交换AX=b中向量 b 的元素
template<typename T>
void swap_vect_row(T* vector, int len, int source_row, int dest_row)
{
	T temp_val = 0;
	temp_val = vector[dest_row - 1];
	vector[dest_row - 1] = vector[source_row - 1];
	vector[source_row - 1] = temp_val;
}

//某行乘以某数
template<typename T>
void mul_row(T* matrix, int row, int col, int source_row, double multiple)
{
	//cout << "第 " << source_row << " 行乘以 " << multiple <<endl;
	for (int i = 0; i < col; i++)
		matrix[(source_row - 1) * col + i] *= multiple;
}

template<typename T>
void add_row(T* matrix, int row, int col, int source_row, int dest_row)
{
	for (int i = 0; i < col; ++i)
		matrix[(dest_row - 1) * col + i] += matrix[(source_row - 1) * col + i];
}

template<typename T>
void add_vec_row(T* vector, int len, int source_row, int dest_row)
{
	vector[dest_row - 1] += vector[source_row - 1];
}

template<typename T>
void mul_vec_row(T* vector, int len, int source_row, double mul)
{
	vector[source_row - 1] *= mul;
}


//矩阵上三角化
//template<typename T,typename I>
void up_triangle(double* matrix, int row, int col, double* vector, int len)  // vector为AX=b中的b
{
	int k = 1;
	double mul_val = 0;
	double* temp_array = new double[row];
	//三角化的过程中是先行再列..
	//这里是从矩阵的视角来看的 所以从第1行开始..
	for (int j = 1; j < col; ++j)  //j为列
	{
		print_matrix_vec((double*)matrix, row, col, vector, len); //打印三角化的过程
		//将主元素换到顺序主子式第一行
		double max_val = 0.0;
		int	   max_row = 0;
		for (int i = j; i <= row; ++i)   //选出在该列中选出最大的行元素作为首元素
		{
			if (fabs(matrix[(i - 1) * col + (j - 1)]) > fabs(max_val))
			{
				max_val = matrix[(i - 1) * col + (j - 1)];
				max_row = i;
			}
		}
		swap_row(matrix, row, col, max_row, j);
		swap_vect_row(vector, len, max_row, j);
		//print_mat((double*)matrix, row, col);   

		//print_matrix_vec((double*)matrix, row, col, vector, len);

		for (int i = j + 1; i <= row; ++i)
		{
			mul_val = matrix[(j - 1) * col + (j - 1)] / matrix[(i - 1) * col + (j - 1)];  //对角线元素是下面元素的倍数
			//cout << mul_val<<endl;
			mul_val *= -1;
			mul_row((double*)matrix, row, col, i, mul_val);
			mul_vec_row(vector, len, i, mul_val);

			add_row((double*)matrix, row, col, j, i);
			add_vec_row(vector, len, j, i);

		}
		print_matrix_vec((double*)matrix, row, col, vector, len);
	}
}

//回带求解x
void solution(double* matrix, int row, int col, double* vector, int len)
{
	if (len != row)
	{
		cout << "input error!" << endl;
		return;
	}
	double mul = 0.0;

	up_triangle((double*)matrix, row, col, vector, len); //上三角化
	double* heap_arry = new double[len];   //存放解
	for (int i = row; i >= 1; --i)			//行
	{
		mul = 1.0 / matrix[(i - 1) * col + (i - 1)];
		mul_row((double*)matrix, row, col, i, mul);
		mul_vec_row(vector, len, i, mul);
		for (int j = row; j > i; --j)		//列
		{
			mul = matrix[(i - 1) * col + (j - 1)];
			vector[i - 1] -= vector[j - 1] * mul;
			matrix[(i - 1) * col + (j - 1)] -= mul * matrix[(j - 1) * col + (j - 1)];
			print_matrix_vec((double*)matrix, row, col, vector, len);
		}
		heap_arry[i - 1] = vector[i - 1];
	}
	for (int k = 0; k < row; k++)
		cout << "x" << k + 1 << " = " << heap_arry[k] << "   ";
	cout << endl;
}


int main()
{
	double matrix[][3] = { {2,2,2},{3,2,4},{1,3,9} };

	double vec[3] = { 1,0.5,2.5 };
	//up_triangle((double*)matrix,3,3,vec,3);
	solution((double*)matrix, 3, 3, vec, 3);
}

输入矩阵:

输出矩阵: