高斯消元法

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高斯消元法是最常用的一种解方程的方式.

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#include <iostream>
#include <iomanip>

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;
	//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)
	{
		print_matrix_vec((double*)matrix, row, col,vector,len); //打印三角化的过程
		//print_mat((double*)matrix, row, col);   
		for (int i = j; i <= row; ++i)
		{
			if (i == j) //当是对角线上的元素
			{
				k = j;
				if (0 == matrix[(i-1) * col + (j-1)])// 当第一个元素为 0 时候需要交换行
				{
					for (k = j + 1; k <= row; ++k) // 遍历matrix[k][1]选出第一个k行不为0的元素
					{
						if (matrix[(k-1) * col + (j-1)] != 0)
							break;
						if (k + 1 == row)  //当该列全为 0
						{
							cout << "矩阵秩为0!" << endl;
							return;
						}
					}
				}
				if (j != k)
				{
					swap_row(matrix, row, col, j, k);	//把非0元素置换
					swap_vect_row(vector, len, j, k);
				}
			}
			else
			{
				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
				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);
			}
		}
	}
}

//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:

高斯消元过程及结果: