C++矩阵运算类（Matrix.h）

这个类数据类型是double，包含了常用的矩阵计算，多数方法经过实践验证，也难免有不足之处，如有发现欢迎指出。

https://github.com/ims0/comTutor/tree/master/matrix

#include <iostream>
#include <fstream>
#include <stdlib.h>
#include <cmath>
using namespace std;
#ifndef _In_opt_#define _In_opt_
#endif
#ifndef _Out_#define _Out_
#endiftypedef unsigned Index_T;
class Matrix
{
private:Index_T m_row, m_col;Index_T m_size;Index_T m_curIndex;double *m_ptr;//数组指针
public:Matrix(Index_T r, Index_T c) :m_row(r), m_col(c)//非方阵构造{m_size = r*c;if (m_size>0){m_ptr = new double[m_size];}elsem_ptr = NULL;};Matrix(Index_T r, Index_T c, double val ) :m_row(r), m_col(c)// 赋初值val{m_size = r*c;if (m_size>0){m_ptr = new double[m_size];}elsem_ptr = NULL;};Matrix(Index_T n) :m_row(n), m_col(n)//方阵构造{m_size = n*n;if (m_size>0){m_ptr = new double[m_size];}elsem_ptr = NULL;};Matrix(const Matrix &rhs)//拷贝构造{m_row = rhs.m_row;m_col = rhs.m_col;m_size = rhs.m_size;m_ptr = new double[m_size];for (Index_T i = 0; i<m_size; i++)m_ptr[i] = rhs.m_ptr[i];}~Matrix(){if (m_ptr != NULL){delete[]m_ptr;m_ptr = NULL;}}Matrix  &operator=(const Matrix&);  //如果类成员有指针必须重写赋值运算符，必须是成员friend istream &operator>>(istream&, Matrix&);friend ofstream &operator<<(ofstream &out, Matrix &obj);  // 输出到文件friend ostream &operator<<(ostream&, Matrix&);          // 输出到屏幕friend Matrix &operator<<(Matrix &mat, const double val);friend Matrix& operator,(Matrix &obj, const double val);friend Matrix  operator+(const Matrix&, const Matrix&);friend Matrix  operator-(const Matrix&, const Matrix&);friend Matrix  operator*(const Matrix&, const Matrix&);  //矩阵乘法friend Matrix  operator*(double, const Matrix&);  //矩阵乘法friend Matrix  operator*(const Matrix&, double);  //矩阵乘法friend Matrix  operator/(const Matrix&, double);  //矩阵 除以单数Matrix multi(const Matrix&); // 对应元素相乘Matrix mtanh(); // 对应元素相乘Index_T row()const{ return m_row; }Index_T col()const{ return m_col; }Matrix getrow(Index_T index); // 返回第index 行,索引从0 算起Matrix getcol(Index_T index); // 返回第index 列Matrix cov(_In_opt_ bool flag = true);   //协方差阵 或者样本方差double det();   //行列式Matrix solveAb(Matrix &obj);  // b是行向量或者列向量Matrix diag();  //返回对角线元素//Matrix asigndiag();  //对角线元素Matrix T()const;   //转置void sort(bool);//true为从小到大Matrix adjoint();Matrix inverse();void QR(_Out_ Matrix&, _Out_ Matrix&)const;Matrix eig_val(_In_opt_ Index_T _iters = 1000);Matrix eig_vect(_In_opt_ Index_T _iters = 1000);double norm1();//1范数double norm2();//2范数double mean();// 矩阵均值double*operator[](Index_T i){ return m_ptr + i*m_col; }//注意this加括号， (*this)[i][j]void zeromean(_In_opt_  bool flag = true);//默认参数为true计算列void normalize(_In_opt_  bool flag = true);//默认参数为true计算列Matrix exponent(double x);//每个元素x次幂Matrix  eye();//对角阵void  maxlimit(double max,double set=0);//对角阵
};/*
类方法的实现
*/Matrix Matrix::mtanh() // 对应元素 tanh()
{Matrix ret(m_row, m_col);for (Index_T i = 0; i<ret.m_size; i++){ret.m_ptr[i] = tanh(m_ptr[i]);}return ret;
}
/** 递归调用*/
double calcDet(Index_T n, double *&aa)
{if (n == 1)return aa[0];double *bb = new double[(n - 1)*(n - 1)];//创建n-1阶的代数余子式阵bbdouble sum = 0.0;for (Index_T Ai = 0; Ai<n; Ai++){for (Index_T Bi = 0; Bi < n - 1; Bi++)//把aa阵第一列各元素的代数余子式存到bb{Index_T offset =  Bi < Ai ? 0 : 1; //bb中小于Ai的行，同行赋值，等于的错过，大于的加一for (Index_T j = 0; j<n - 1; j++){bb[Bi*(n - 1) + j] = aa[(Bi + offset)*n + j + 1];}}int flag = (Ai % 2 == 0 ? 1 : -1);//因为列数为0，所以行数是偶数时候，代数余子式为1.sum += flag* aa[Ai*n] * calcDet(n - 1, bb);//aa第一列各元素与其代数余子式积的和即为行列式}delete[]bb;return sum;
}Matrix Matrix::solveAb(Matrix &obj)
{Matrix ret(m_row, 1);if (m_size == 0 || obj.m_size == 0){cout << "solveAb(Matrix &obj):this or obj is null" << endl;return ret;}if (m_row != obj.m_size){cout << "solveAb(Matrix &obj):the row of two matrix is not equal!" << endl;return ret;}double *Dx = new double[m_row*m_row];for (Index_T i = 0; i<m_row; i++){for (Index_T j = 0; j<m_row; j++){Dx[i*m_row + j] = m_ptr[i*m_row + j];}}double D = calcDet(m_row, Dx);if (D == 0){cout << "Cramer法则只能计算系数矩阵为满秩的矩阵" << endl;return  ret;}for (Index_T j = 0; j<m_row; j++){for (Index_T i = 0; i<m_row; i++){for (Index_T j = 0; j<m_row; j++){Dx[i*m_row + j] = m_ptr[i*m_row + j];}}for (Index_T i = 0; i<m_row; i++){Dx[i*m_row + j] = obj.m_ptr[i]; //obj赋值给第j列}//for( int i=0;i<m_row;i++) //print//{//    for(int j=0; j<m_row;j++)//    {//        cout<< Dx[i*m_row+j]<<"\t";//    }//    cout<<endl;//}ret[j][0] = calcDet(m_row, Dx) / D;}delete[]Dx;return ret;
}Matrix Matrix::getrow(Index_T index)//返回行
{Matrix ret(1, m_col); //一行的返回值for (Index_T i = 0; i< m_col; i++){ret[0][i] = m_ptr[(index) *m_col + i] ;}return ret;
}Matrix Matrix::getcol(Index_T index)//返回列
{Matrix ret(m_row, 1); //一列的返回值for (Index_T i = 0; i< m_row; i++){ret[i][0] = m_ptr[i *m_col + index];}return ret;
}Matrix Matrix::exponent(double x)//每个元素x次幂
{Matrix ret(m_row, m_col);for (Index_T i = 0; i< m_row; i++){for (Index_T j = 0; j < m_col; j++){ret[i][j]= pow(m_ptr[i*m_col + j],x);}}return ret;
}
void Matrix::maxlimit(double max, double set)//每个元素x次幂
{for (Index_T i = 0; i< m_row; i++){for (Index_T j = 0; j < m_col; j++){m_ptr[i*m_col + j] = m_ptr[i*m_col + j]>max ? 0 : m_ptr[i*m_col + j];}}}
Matrix Matrix::eye()//对角阵
{for (Index_T i = 0; i< m_row; i++){for (Index_T j = 0; j < m_col; j++){if (i == j){m_ptr[i*m_col + j] = 1.0;}}}return *this;
}
void Matrix::zeromean(_In_opt_  bool flag)
{if (flag == true) //计算列均值{double *mean = new double[m_col];for (Index_T j = 0; j < m_col; j++){mean[j] = 0.0;for (Index_T i = 0; i < m_row; i++){mean[j] += m_ptr[i*m_col + j];}mean[j] /= m_row;}for (Index_T j = 0; j < m_col; j++){for (Index_T i = 0; i < m_row; i++){m_ptr[i*m_col + j] -= mean[j];}}delete[]mean;}else //计算行均值{double *mean = new double[m_row];for (Index_T i = 0; i< m_row; i++){mean[i] = 0.0;for (Index_T j = 0; j < m_col; j++){mean[i] += m_ptr[i*m_col + j];}mean[i] /= m_col;}for (Index_T i = 0; i < m_row; i++){for (Index_T j = 0; j < m_col; j++){m_ptr[i*m_col + j] -= mean[i];}}delete[]mean;}
}void Matrix::normalize(_In_opt_  bool flag)
{if (flag == true) //计算列均值{double *mean = new double[m_col];for (Index_T j = 0; j < m_col; j++){mean[j] = 0.0;for (Index_T i = 0; i < m_row; i++){mean[j] += m_ptr[i*m_col + j];}mean[j] /= m_row;}for (Index_T j = 0; j < m_col; j++){for (Index_T i = 0; i < m_row; i++){m_ptr[i*m_col + j] -= mean[j];}}///计算标准差for (Index_T j = 0; j < m_col; j++){mean[j] = 0;for (Index_T i = 0; i < m_row; i++){mean[j] += pow(m_ptr[i*m_col + j],2);//列平方和}mean[j] = sqrt(mean[j] / m_row); // 开方}for (Index_T j = 0; j < m_col; j++){for (Index_T i = 0; i < m_row; i++){m_ptr[i*m_col + j] /= mean[j];//列平方和}}delete[]mean;}else //计算行均值{double *mean = new double[m_row];for (Index_T i = 0; i< m_row; i++){mean[i] = 0.0;for (Index_T j = 0; j < m_col; j++){mean[i] += m_ptr[i*m_col + j];}mean[i] /= m_col;}for (Index_T i = 0; i < m_row; i++){for (Index_T j = 0; j < m_col; j++){m_ptr[i*m_col + j] -= mean[i];}}///计算标准差for (Index_T i = 0; i< m_row; i++){mean[i] = 0.0;for (Index_T j = 0; j < m_col; j++){mean[i] += pow(m_ptr[i*m_col + j], 2);//列平方和}mean[i] = sqrt(mean[i] / m_col); // 开方}for (Index_T i = 0; i < m_row; i++){for (Index_T j = 0; j < m_col; j++){m_ptr[i*m_col + j] /= mean[i];}}delete[]mean;}
}double Matrix::det()
{if (m_col == m_row)return calcDet(m_row, m_ptr);else{cout << ("行列不相等无法计算") << endl;return 0;}
}
/
istream& operator>>(istream &is, Matrix &obj)
{for (Index_T i = 0; i<obj.m_size; i++){is >> obj.m_ptr[i];}return is;
}ostream& operator<<(ostream &out, Matrix &obj)
{for (Index_T i = 0; i < obj.m_row; i++) //打印逆矩阵{for (Index_T j = 0; j < obj.m_col; j++){out << (obj[i][j]) << "\t";}out << endl;}return out;
}
ofstream& operator<<(ofstream &out, Matrix &obj)//打印逆矩阵到文件
{for (Index_T i = 0; i < obj.m_row; i++){for (Index_T j = 0; j < obj.m_col; j++){out << (obj[i][j]) << "\t";}out << endl;}return out;
}Matrix& operator<<(Matrix &obj, const double val)
{*obj.m_ptr = val;obj.m_curIndex = 1;return obj;
}
Matrix& operator,(Matrix &obj, const double val)
{if( obj.m_curIndex == 0 || obj.m_curIndex > obj.m_size - 1 ){return obj;}*(obj.m_ptr + obj.m_curIndex) = val;++obj.m_curIndex;return obj;
}Matrix operator+(const Matrix& lm, const Matrix& rm)
{if (lm.m_col != rm.m_col || lm.m_row != rm.m_row){Matrix temp(0, 0);temp.m_ptr = NULL;cout << "operator+(): 矩阵shape 不合适,m_col:"<< lm.m_col << "," << rm.m_col << ".  m_row:" << lm.m_row << ", " << rm.m_row << endl;return temp; //数据不合法时候，返回空矩阵}Matrix ret(lm.m_row, lm.m_col);for (Index_T i = 0; i<ret.m_size; i++){ret.m_ptr[i] = lm.m_ptr[i] + rm.m_ptr[i];}return ret;
}
Matrix operator-(const Matrix& lm, const Matrix& rm)
{if (lm.m_col != rm.m_col || lm.m_row != rm.m_row){Matrix temp(0, 0);temp.m_ptr = NULL;cout << "operator-(): 矩阵shape 不合适,m_col:"<<lm.m_col<<","<<rm.m_col<<".  m_row:"<< lm.m_row <<", "<< rm.m_row << endl;return temp; //数据不合法时候，返回空矩阵}Matrix ret(lm.m_row, lm.m_col);for (Index_T i = 0; i<ret.m_size; i++){ret.m_ptr[i] = lm.m_ptr[i] - rm.m_ptr[i];}return ret;
}
Matrix operator*(const Matrix& lm, const Matrix& rm)  //矩阵乘法
{if (lm.m_size == 0 || rm.m_size == 0 || lm.m_col != rm.m_row){Matrix temp(0, 0);temp.m_ptr = NULL;cout << "operator*(): 矩阵shape 不合适,m_col:"<< lm.m_col << "," << rm.m_col << ".  m_row:" << lm.m_row << ", " << rm.m_row << endl;return temp; //数据不合法时候，返回空矩阵}Matrix ret(lm.m_row, rm.m_col);for (Index_T i = 0; i<lm.m_row; i++){for (Index_T j = 0; j< rm.m_col; j++){for (Index_T k = 0; k< lm.m_col; k++)//lm.m_col == rm.m_row{ret.m_ptr[i*rm.m_col + j] += lm.m_ptr[i*lm.m_col + k] * rm.m_ptr[k*rm.m_col + j];}}}return ret;
}
Matrix operator*(double val, const Matrix& rm)  //矩阵乘 单数
{Matrix ret(rm.m_row, rm.m_col);for (Index_T i = 0; i<ret.m_size; i++){ret.m_ptr[i] = val * rm.m_ptr[i];}return ret;
}
Matrix operator*(const Matrix&lm, double val)  //矩阵乘 单数
{Matrix ret(lm.m_row, lm.m_col);for (Index_T i = 0; i<ret.m_size; i++){ret.m_ptr[i] = val * lm.m_ptr[i];}return ret;
}Matrix operator/(const Matrix&lm, double val)  //矩阵除以 单数
{Matrix ret(lm.m_row, lm.m_col);for (Index_T i = 0; i<ret.m_size; i++){ret.m_ptr[i] =  lm.m_ptr[i]/val;}return ret;
}
Matrix Matrix::multi(const Matrix&rm)// 对应元素相乘
{if (m_col != rm.m_col || m_row != rm.m_row){Matrix temp(0, 0);temp.m_ptr = NULL;cout << "multi(const Matrix&rm): 矩阵shape 不合适,m_col:"<< m_col << "," << rm.m_col << ".  m_row:" << m_row << ", " << rm.m_row << endl;return temp; //数据不合法时候，返回空矩阵}Matrix ret(m_row,m_col);for (Index_T i = 0; i<ret.m_size; i++){ret.m_ptr[i] = m_ptr[i] * rm.m_ptr[i];}return ret;}Matrix&  Matrix::operator=(const Matrix& rhs)
{if (this != &rhs){m_row = rhs.m_row;m_col = rhs.m_col;m_size = rhs.m_size;if (m_ptr != NULL)delete[] m_ptr;m_ptr = new double[m_size];for (Index_T i = 0; i<m_size; i++){m_ptr[i] = rhs.m_ptr[i];}}return *this;
}
//||matrix||_2  求A矩阵的2范数
double Matrix::norm2()
{double norm = 0;for (Index_T i = 0; i < m_size; ++i){norm += m_ptr[i] * m_ptr[i];}return (double)sqrt(norm);
}
double Matrix::norm1()
{double sum = 0;for (Index_T i = 0; i < m_size; ++i){sum += abs(m_ptr[i]);}return sum;
}
double Matrix::mean()
{double sum = 0;for (Index_T i = 0; i < m_size; ++i){sum += (m_ptr[i]);}return sum/m_size;
}void Matrix::sort(bool flag)
{double tem;for (Index_T i = 0; i<m_size; i++){for (Index_T j = i + 1; j<m_size; j++){if (flag == true){if (m_ptr[i]>m_ptr[j]){tem = m_ptr[i];m_ptr[i] = m_ptr[j];m_ptr[j] = tem;}}else{if (m_ptr[i]<m_ptr[j]){tem = m_ptr[i];m_ptr[i] = m_ptr[j];m_ptr[j] = tem;}}}}
}
Matrix Matrix::diag()
{if (m_row != m_col){Matrix m(0);cout << "diag():m_row != m_col" << endl;return m;}Matrix m(m_row);for (Index_T i = 0; i<m_row; i++){m.m_ptr[i*m_row + i] = m_ptr[i*m_row + i];}return m;
}
Matrix Matrix::T()const
{Matrix tem(m_col, m_row);for (Index_T i = 0; i<m_row; i++){for (Index_T j = 0; j<m_col; j++){tem[j][i] = m_ptr[i*m_col + j];// (*this)[i][j]}}return tem;
}
void  Matrix::QR(Matrix &Q, Matrix &R) const
{//如果A不是一个二维方阵，则提示错误，函数计算结束if (m_row != m_col){printf("ERROE: QR() parameter A is not a square matrix!\n");return;}const Index_T N = m_row;double *a = new double[N];double *b = new double[N];for (Index_T j = 0; j < N; ++j)  //(Gram-Schmidt) 正交化方法{for (Index_T i = 0; i < N; ++i)  //第j列的数据存到a，ba[i] = b[i] = m_ptr[i * N + j];for (Index_T i = 0; i<j; ++i)  //第j列之前的列{R.m_ptr[i * N + j] = 0;  //for (Index_T m = 0; m < N; ++m){R.m_ptr[i * N + j] += a[m] * Q.m_ptr[m *N + i]; //R[i,j]值为Q第i列与A的j列的内积}for (Index_T m = 0; m < N; ++m){b[m] -= R.m_ptr[i * N + j] * Q.m_ptr[m * N + i]; //}}double norm = 0;for (Index_T i = 0; i < N; ++i){norm += b[i] * b[i];}norm = (double)sqrt(norm);R.m_ptr[j*N + j] = norm; //向量b[]的2范数存到R[j,j]for (Index_T i = 0; i < N; ++i){Q.m_ptr[i * N + j] = b[i] / norm; //Q 阵的第j列为单位化的b[]}}delete[]a;delete[]b;
}
Matrix Matrix::eig_val(_In_opt_ Index_T _iters)
{if (m_size == 0 || m_row != m_col){cout << "矩阵为空或者非方阵！" << endl;Matrix rets(0);return rets;}//if (det() == 0)//{//  cout << "非满秩矩阵没法用QR分解计算特征值！" << endl;//  Matrix rets(0);//  return rets;//}const Index_T N = m_row;Matrix matcopy(*this);//备份矩阵Matrix Q(N), R(N);/*当迭代次数足够多时,A 趋于上三角矩阵，上三角矩阵的对角元就是A的全部特征值。*/for (Index_T k = 0; k < _iters; ++k){//cout<<"this:\n"<<*this<<endl;QR(Q, R);*this = R*Q;/*  cout<<"Q:\n"<<Q<<endl;cout<<"R:\n"<<R<<endl;  */}Matrix val = diag();*this = matcopy;//恢复原始矩阵；return val;
}
Matrix Matrix::eig_vect(_In_opt_ Index_T _iters)
{if (m_size == 0 || m_row != m_col){cout << "矩阵为空或者非方阵！" << endl;Matrix rets(0);return rets;}if (det() == 0){cout << "非满秩矩阵没法用QR分解计算特征向量！" << endl;Matrix rets(0);return rets;}Matrix matcopy(*this);//备份矩阵Matrix eigenValue = eig_val(_iters);Matrix ret(m_row);const Index_T NUM = m_col;double eValue;double sum, midSum, diag;Matrix copym(*this);for (Index_T count = 0; count < NUM; ++count){//计算特征值为eValue，求解特征向量时的系数矩阵*this = copym;eValue = eigenValue[count][count];for (Index_T i = 0; i < m_col; ++i)//A-lambda*I{m_ptr[i * m_col + i] -= eValue;}//cout<<*this<<endl;//将 this为阶梯型的上三角矩阵for (Index_T i = 0; i < m_row - 1; ++i){diag = m_ptr[i*m_col + i];  //提取对角元素for (Index_T j = i; j < m_col; ++j){m_ptr[i*m_col + j] /= diag; //【i,i】元素变为1}for (Index_T j = i + 1; j<m_row; ++j){diag = m_ptr[j *  m_col + i];for (Index_T q = i; q < m_col; ++q)//消去第i+1行的第i个元素{m_ptr[j*m_col + q] -= diag*m_ptr[i*m_col + q];}}}//cout<<*this<<endl;//特征向量最后一行元素置为1midSum = ret.m_ptr[(ret.m_row - 1) * ret.m_col + count] = 1;for (int m = m_row - 2; m >= 0; --m){sum = 0;for (Index_T j = m + 1; j < m_col; ++j){sum += m_ptr[m *  m_col + j] * ret.m_ptr[j * ret.m_col + count];}sum = -sum / m_ptr[m *  m_col + m];midSum += sum * sum;ret.m_ptr[m * ret.m_col + count] = sum;}midSum = sqrt(midSum);for (Index_T i = 0; i < ret.m_row; ++i){ret.m_ptr[i * ret.m_col + count] /= midSum; //每次求出一个列向量}}*this = matcopy;//恢复原始矩阵；return ret;
}
Matrix Matrix::cov(bool flag)
{//m_row 样本数，column 变量数if (m_col == 0){Matrix m(0);return m;}double *mean = new double[m_col]; //均值向量for (Index_T j = 0; j<m_col; j++) //init{mean[j] = 0.0;}Matrix ret(m_col);for (Index_T j = 0; j<m_col; j++) //mean{for (Index_T i = 0; i<m_row; i++){mean[j] += m_ptr[i*m_col + j];}mean[j] /= m_row;}Index_T i, k, j;for (i = 0; i<m_col; i++) //第一个变量{for (j = i; j<m_col; j++) //第二个变量{for (k = 0; k<m_row; k++) //计算{ret[i][j] += (m_ptr[k*m_col + i] - mean[i])*(m_ptr[k*m_col + j] - mean[j]);}if (flag == true){ret[i][j] /= (m_row-1);}else{ret[i][j] /= (m_row);}}}for (i = 0; i<m_col; i++) //补全对应面{for (j = 0; j<i; j++){ret[i][j] = ret[j][i];}}return ret;
}/** 返回代数余子式*/
double CalcAlgebraicCofactor( Matrix& srcMat, Index_T ai, Index_T aj)
{Index_T temMatLen = srcMat.row()-1;Matrix temMat(temMatLen);for (Index_T bi = 0; bi < temMatLen; bi++){for (Index_T bj = 0; bj < temMatLen; bj++){Index_T rowOffset = bi < ai ? 0 : 1;Index_T colOffset = bj < aj ? 0 : 1;temMat[bi][bj] = srcMat[bi + rowOffset][bj + colOffset];}}int flag = (ai + aj) % 2 == 0 ? 1 : -1;return flag * temMat.det();
}/** 返回伴随阵*/
Matrix Matrix::adjoint()
{if (m_row != m_col){return Matrix(0);}Matrix adjointMat(m_row);for (Index_T ai = 0; ai < m_row; ai++){for (Index_T aj = 0; aj < m_row; aj++){adjointMat.m_ptr[aj*m_row + ai] = CalcAlgebraicCofactor(*this, ai, aj);}}return adjointMat;
}Matrix Matrix::inverse()
{double detOfMat = det();if (detOfMat == 0){cout << "行列式为0，不能计算逆矩阵。" << endl;return Matrix(0);}return adjoint()/detOfMat;
}

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