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Copy pathrematrix.cpp
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801 lines (739 loc) · 24.8 KB
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//
// Created by A Luck Boy on 2023/4/24.
//
#include "rematrix.h"
Vector::Vector() {
this->length = 1;
this->data = new Element[1];
this->data[0] = 0.;
}
Vector::Vector(Element* _data, int _length) {
this->length = _length;
this->data = new Element[_length]();
for (int i = 0; i < length; i++) {
this->data[i] = _data[i];
}
}
// [BugFix #1] 析构函数必须释放 data，否则内存泄漏
Vector::~Vector() {
delete[] data;
}
Vector::Vector(const Vector &other) {
this->length = other.length;
data = new Element[length];
for (int i = 0; i < length; i++) {
data[i] = other.data[i];
}
}
// [BugFix #2] 赋值运算符深拷贝，防止浅拷贝导致 double-free / 数据串改
Vector& Vector::operator=(const Vector &other) {
if (this == &other) return *this;
delete[] data;
this->length = other.length;
data = new Element[length];
for (int i = 0; i < length; i++) {
data[i] = other.data[i];
}
return *this;
}
Vector Vector::expand(int index, double k) {
if (index < 0 || index > length-1){
error_models(9);
exit(EXIT_FAILURE);
}
auto _data = new Element [length];
for (int i=0;i<length;i++){ _data[i] = data[i];}
_data[index] = _data[index] * k;
auto vec = Vector(_data, length);
delete[] _data;
return vec;
}
Vector * Vector::transpose() {
auto* vecs = new Vector[length];
for (int i = 0; i < length; i++) {
Vector temp = Vector();
temp.data[0] = data[i];
vecs[i] = temp;
}
return vecs;
}
bool Vector::isEqual(Vector &other) const {
if (other.length == this->length){
for (int i=0;i< this->length;i++){
if (this->data[i] != other.data[i]) return false;
}
return true;
}
return false;
}
// [BugFix #3] 原实现使用默认构造 Vector()（只分配1个元素）再写入length个元素 → 越界
Vector Vector::multi(Element k) {
auto _data = new Element[length];
for (int i = 0; i < length; i++) {
_data[i] = data[i] * k;
}
auto vec = Vector(_data, length);
delete[] _data;
return vec;
}
Vector Vector::add(Vector &other) {
if (other.length != this->length){
error_models(8);
exit(EXIT_FAILURE);
}
auto vec_data = new Element[this->length];
for (int i=0;i<length;i++){
vec_data[i] = this->data[i] + other.data[i];
}
auto vec = Vector(vec_data, length);
delete[] vec_data;
return vec;
}
bool Vector::isZero() const {
int _count = 0;
for(int i=0;i<length;i++) {if(data[i] == 0) ++_count; }
return _count == length;
}
Vector Vector::unit() {
if (isZero()) { error_models(0);exit(EXIT_FAILURE); }
auto datas = new Element[length];
for (int i=0;i<length;i++){ datas[i] = this->data[i]* this->data[i];}
Element mod = pow(sums(datas, length), .5);
delete[] datas;
auto _dat = new Element [length];
for (int i=0;i<length;i++){
_dat[i] = this->data[i] / mod;
}
auto vec = Vector(_dat, length);
delete[] _dat;
return vec;
}
Element Vector::inner(Vector &other) const{
if (this->length != other.length) { error_models(8);exit(EXIT_FAILURE); }
Element e = 0.;
for (int i=0;i<length;i++){
e += this->data[i] * other.data[i];
}
return e;
}
Vector transpose2(Vector *vec, int row){
auto data = new Element[row];
for (int _row=0;_row<row;_row++){
data[_row] = vec[_row].data[0];
}
auto _vec = Vector(data, row);
delete[] data;
return _vec;
}
void Vector::printVec(int len ,bool ifself, int round) {
if (ifself) { cout << "\t\033[1;32m[ "; }
else {cout << "\n\t\033[1;32m[ "; }
for (int i = 0; i < len - 1; i++) {
cout << fixed << setprecision(round)<< data[i] << " ";
}
cout << fixed << setprecision(round)<< data[len - 1] << " ";
cout << fixed << setprecision(ROUND);
cout << "\033[1;32m]";
}
void printVecC(Vector* vecs, int length, int round){
cout << "\t\033[1;32m[ ";
for (int i = 0; i < length; i++) {
cout << "[ " << fixed << setprecision(round)<< vecs[i].data[0] << " \033[1;32m] ";
}
cout << fixed << setprecision(ROUND);
cout << "\033[1;32m]";
}
Vector Vector::splice(Vector &other, bool pos) {
auto _data = new Element[this->length + other.length];
if (pos){
for (int i=0;i< this->length;i++) _data[i] = this->data[i];
for (int i=0;i< other.length;i++) _data[i+ this->length] = other.data[i];
}
else{
for (int i=0;i< other.length;i++) _data[i] = other.data[i];
for (int i=0;i< this->length;i++) _data[i+other.length] = this->data[i];
}
auto vec = Vector(_data, this->length+other.length);
delete[] _data;
return vec;
}
int getReverseOrderNumber(Vector & vec) {
int _count = 0;
for (int i=0;i< vec.length-1;i++){
for (int j=i+1;j<vec.length;j++){ if(vec.data[i] > vec.data[j]) ++_count; } }
return _count;
}
// The core codes of the Matrix
Matrix::Matrix(Vector * vectors, int _row, int _column){
this->shape = {_row, _column};
this->_row = _row;
this->_column = _column;
for (int row=0;row<this->_row;row++) {
this->_vecs[row] = vectors[row];
this->_vecs[row].length = _column;
}
this->spc = {false, false, false, false, false, false};
}
Matrix::Matrix(const Matrix &other) {
this->_row = other._row;
this->_column = other._column;
for (int i=0;i< this->_row;i++)
{
this->_vecs[i] = other._vecs[i];
this->_vecs[i].length = other._vecs[i].length;
}
this->shape.row = other.shape.row;
this->shape.column = other.shape.column;
this->spc.digUp = other.spc.digUp;
this->spc.digDown = other.spc.digDown;
this->spc.triDown = other.spc.triDown;
this->spc.triUp = other.spc.triUp;
this->spc.unit = other.spc.unit;
this->spc.symmetry = other.spc.symmetry;
}
Matrix::~Matrix() = default;
void Matrix::printMatrix(int round) {
cout << "\t============当前矩阵============";
for (int vec=0;vec< this->_row;vec++){
this->_vecs[vec].printVec(this->_column, false, round);
}
cout << "\n\t===============================";
}
bool Matrix::isSquare() const { return _row == _column; }
Matrix Matrix::E() {
if (!isSquare()) { error_models(4); exit(EXIT_FAILURE); }
auto vecs = new Vector[this->_row];
for (int i=0;i< this->_column;i++){
auto e_vector = new Element[_row];
for (int j=0;j!=_row;j++){
e_vector[j] = 0;
}
e_vector[i] = 1;
vecs[i] = Vector(e_vector, _row);
delete[] e_vector;
}
auto e = Matrix(vecs, _row, _row);
e.spc = {true, true, true, false, true, true};
return e;
}
Matrix Matrix::transpose() {
if (this->spc.unit) { return copy_mat(*this);}
if (this->spc.symmetry) {return copy_mat(*this); }
auto vecs = new Vector[this->_column];
for (int i=0;i< this->_column;i++){
auto data = new Element[this->_column];
for (int dat=0;dat< this->_column;dat++){
data[dat] = this->_vecs[dat].data[i];
}
vecs[i] = Vector(data, this->_row);
delete[] data;
}
return Matrix(vecs, this->_column, this->_row);
}
bool Matrix::isDig(bool if_main) {
if (!isSquare()) { error_models(4);exit(EXIT_FAILURE) ;}
if (if_main){
if (this->spc.unit) {
this->spc.digUp = true;
return true;
}
if (this->spc.symmetry) {
this->spc.digUp = true;
return true;
}
for (int i=0;i< this->_row;i++){
for (int j=0;j< this->_column;j++){
if (i!=j) {
if ( this->_vecs[i].data[j] != 0) return false;
}
}
}
this->spc.digUp = true;
return true;
}
else{
for (int i=0;i< this->_row;i++){
for (int j=0;j< this->_column;j++){
if (i != this->_row-j-1){
if (this->_vecs[i].data[j] != 0) return false;
}
}
}
}
this->spc.digDown = true;
return true;
}
bool Matrix::isSymmetric() {
if(!isSquare()) { error_models(4);exit(EXIT_FAILURE) ;}
if (this->spc.unit) {
this->spc.symmetry = true;
return true;
}
for (int i=0;i< this->_row;i++){
for (int j=0;j< this->_column;j++)
{
if (this->_vecs[i].data[j] != this->_vecs[j].data[i]) return false;
}
}
this->spc.symmetry = true;
return true;
}
// [BugFix #4] 修复两处 bug:
// a) 边界检查 "row > this->_column-1" 应为 "col > this->_column-1"
// b) 构造子矩阵的 Vector 长度应为 this->_column-1 而非 this->_column（否则越界读取）
Matrix Matrix::getRemainder(int row, int col) {
if (this->_row <= 1 || this->_column <= 1) { error_models(6);exit(EXIT_FAILURE); }
if ((row < 0 || row > this->_row-1) || (col < 0 || col > this->_column-1))
{
error_models(6);
exit(EXIT_FAILURE);
}
auto vecs = new Vector[this->_row-1];
int index = 0;
for (int r=0;r< this->_row;r++){
int index2 = 0;
if (r != row){
auto data = new Element[this->_column-1];
for (int c=0;c< this->_column;c++){
if (c != col) {
data[index2] = this->_vecs[r].data[c];
index2++;
}
}
auto temp = Vector(data, this->_column - 1);
delete[] data;
vecs[index] = temp;
index++;
}
}
return Matrix(vecs, _row-1, _column-1);
}
Matrix Matrix::multi(Element k) {
auto vecs = new Vector[this->_row];
for (int i=0;i< this->_row;i++)
{
vecs[i] = this->_vecs[i].multi(k);
}
return Matrix(vecs, _row, _column);
}
Matrix Matrix::operator * (Element k) { return multi(k); }
Matrix Matrix::add(Matrix &other) {
if (other._column == this->_column && other._row == this->_row){
auto vecs = new Vector[this->_row];
for (int i=0;i< this->_row;i++){
vecs[i] = this->_vecs[i].add(other._vecs[i]);
}
return Matrix(vecs, this->_row, this->_column);
} else{
error_models(1);
exit(EXIT_FAILURE);
}
}
Matrix Matrix::operator+( Matrix &other) {return add(other);}
Matrix Matrix::splice(Matrix &other, Splice _splice) {
// 如果左右拼接
if (_splice.if_lr){
if (this->_row != other._row) { error_models(2); exit(EXIT_FAILURE); }
auto vesc = new Vector[this->_row];
// 放other于右侧
if (_splice.other_lr){
for (int i=0;i< this->_row;i++){ vesc[i] = this->_vecs[i].splice(other._vecs[i]); }
}
else { for (int i=0;i< this->_row;i++)vesc[i] = other._vecs[i].splice(this->_vecs[i]);}
return Matrix(vesc, this->_row, this->_column+ other._column);
}
// 如果上下拼接
else
{
if (this->_column != other._column) { error_models(3);exit(EXIT_FAILURE);}
auto vesc = new Vector[this->_row+other._row];
// 放other于下面
if (_splice.other_lr) {
for(int r=0;r< this->_row;r++) vesc[r] = this->_vecs[r];
for(int r=0;r< other._row;r++) vesc[r+ this->_row] = other._vecs[r];
}
else{
for(int r=0;r< other._row;r++) vesc[r] = other._vecs[r];
for(int r=0;r< this->_row;r++) vesc[r+ other._row] = this->_vecs[r];
}
return Matrix(vesc, this->_row+ other._row, this->_column);
}
}
Matrix Matrix::operator^(Matrix &other) { return innerMulti(other);}
bool Matrix::isEqual(Matrix &other) {
if (other._row == this->_row && other._column == this->_column){
for (int i=0;i< this->_row;i++){
if (!this->_vecs[i].isEqual(other._vecs[i])) return false;
}
this->spc.symmetry = other.spc.symmetry = (this->spc.symmetry || other.spc.symmetry);
this->spc.digUp = other.spc.digUp = (this->spc.digUp || other.spc.digUp);
this->spc.digDown = other.spc.digDown = (this->spc.digDown || other.spc.digDown);
this->spc.triDown = other.spc.triDown = (this->spc.triDown || other.spc.triDown);
this->spc.triUp = other.spc.triUp = (this->spc.triUp || other.spc.triUp);
this->spc.unit = other.spc.unit = (this->spc.unit || other.spc.unit);
return true;
}
return false;
}
bool Matrix::operator==(Matrix &other) { return isEqual(other);}
// [BugFix #5] 下三角判断的 else 分支误写 spc.triUp = true，应为 spc.triDown = true
bool Matrix::ifTriMatrix(bool if_up) {
if (!isSquare()) { error_models(4);exit(EXIT_FAILURE); }
if (this->spc.unit){
this->spc.triUp = true;
this->spc.triDown = true;
return true;
}
if (if_up){
for (int i=1;i< this->_row;i++) {
for (int j = 0; j < i; j++) {
if (this->_vecs[i].data[j] != 0)return false;
}
}
this->spc.triUp = true;
return true;
}
else{
for (int i=0;i< this->_row;++i){
for (int j=i+1;j< this->_row;j++){
if(this->_vecs[i].data[j] != 0)return false;
}
}
this->spc.triDown = true; // ← 修复: 原为 triUp
return true;
}
}
void Matrix::getElementaryTransposeReSize(int lr, double k, bool if_line) {
if (if_line){
if(lr < 0 || lr > this->_row-1){
error_models(6);
exit(EXIT_FAILURE);
}
this->_vecs[lr] = this->_vecs[lr].multi(k);
}
else
{
if(lr <0 || lr > this->_column-1){
error_models(6);
exit(EXIT_FAILURE);
}
for (int j=0;j< this->_row;j++){
this->_vecs[j] = this->_vecs[j].expand(lr, k);
}
}
}
void Matrix::getElementaryTransposeExchange(int lr1, int lr2, bool if_line) {
if (if_line){
if (lr1 < 0 || lr2 < 0|| lr1 > this->_row-1||lr2 > this->_row-1){
error_models(6) ;
exit(EXIT_FAILURE);
}
auto temp = this->_vecs[lr1];
this->_vecs[lr1] = this->_vecs[lr2];
this->_vecs[lr2] = temp;
} else{
if(lr1 < 0||lr2 < 0|| lr1 > this->_column-1 || lr2 > this->_column-1){
error_models(6) ;
exit(EXIT_FAILURE);
}
for (int j=0;j< this->_row;j++){
auto temp = this->_vecs[j].data[lr1];
this->_vecs[j].data[lr1] = this->_vecs[j].data[lr2];
this->_vecs[j].data[lr2] = temp;
}
}
}
void Matrix::getElementaryTransposAdd(int lr1, int lr2, double k, bool if_line) {
if (if_line){
if (lr2 <0 || lr1 < 0 || lr1 > this->_row-1 || lr2 > this->_row-1){
error_models(6) ;
exit(EXIT_FAILURE);
}
auto _vec = this->_vecs[lr2].multi(k);
this->_vecs[lr1] = this->_vecs[lr1].add(_vec);
} else{
if (lr2 <0 || lr1 < 0 || lr1 > this->_column-1 || lr2 > this->_column-1){
error_models(6) ;
exit(EXIT_FAILURE);
}
for (int j=0;j< this->_row;j++){
this->_vecs[j].data[lr1] = this->_vecs[j].data[lr1] + this->_vecs[j].data[lr2] * k;
}
}
}
// [优化] 行列式计算
// 保留对已知特殊矩阵（单位阵、三角阵、对角阵）的 O(1) 快速路径
// 一般情况改为高斯消元（带列主元），O(n³) 替代原 O(n!) 递归代数余子式
Element Matrix::det(){
if (!isSquare()) { error_models(4);exit(EXIT_FAILURE); }
if (this->spc.unit) {return 1;}
elif (this->spc.triUp || this->spc.triDown || this->spc.digUp) {
Element _sum = 1;
for (int i=0;i< this->_row;i++)
{
_sum *= this->_vecs[i].data[i] ;
}return _sum;
}
elif(this->spc.digDown){
// [BugFix #6] 用整数运算计算符号，避免 pow(-1, 浮点数) 产生 NaN
Element _sum = 1;
for (int i=0;i< this->_row;i++)
{
_sum *= this->_vecs[i].data[this->_column-i-1] ;
}
int n = this->_row;
int sign = ((n * (n - 1) / 2) % 2 == 0) ? 1 : -1;
return _sum * sign;
}
// 一般情况: 高斯消元法（带部分列主元），O(n³)
else{
int n = this->_row;
// 创建工作矩阵（栈上分配，n <= MAXCOUNT = 10）
Element mat[MAXCOUNT][MAXCOUNT];
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
mat[i][j] = this->_vecs[i].data[j];
}
}
Element det_sign = 1;
for (int col = 0; col < n; col++) {
// 部分列主元: 选当前列中绝对值最大的行
int max_row = col;
Element max_val = fabs(mat[col][col]);
for (int row = col + 1; row < n; row++) {
if (fabs(mat[row][col]) > max_val) {
max_val = fabs(mat[row][col]);
max_row = row;
}
}
// 主元为零 → 奇异矩阵，行列式为 0
if (max_val < EPSILON) {
return 0;
}
// 交换行（记录符号变化）
if (max_row != col) {
for (int j = 0; j < n; j++) {
Element temp = mat[col][j];
mat[col][j] = mat[max_row][j];
mat[max_row][j] = temp;
}
det_sign *= -1;
}
// 消元: 将当前列 col 下方元素消为 0
for (int row = col + 1; row < n; row++) {
Element factor = mat[row][col] / mat[col][col];
for (int j = col; j < n; j++) {
mat[row][j] -= factor * mat[col][j];
}
}
}
// 行列式 = 符号 × 对角元素乘积
Element result = det_sign;
for (int i = 0; i < n; i++) {
result *= mat[i][i];
}
return result;
}
}
// [BugFix #7] 使用 EPSILON 进行浮点比较，避免因精度问题误判
bool Matrix::isSingularMat() { return fabs(det()) < EPSILON;}
Matrix Matrix::Zero(){
auto _like_this = copy_mat(*this);
// 这一步可以防止spc被继承之和，参与一些检测方法引发歧义
_like_this.spc = {false, false, false, false, false, false};
return _like_this * 0;
}
Matrix Matrix::getAccompany() {
if (!isSquare()){ error_models(4); exit(EXIT_FAILURE); }
if (isSingularMat()) { error_models(10);exit(EXIT_FAILURE);}
elif (this->spc.unit) {return copy_mat(*this);}
elif (this->spc.digUp){
auto _mat = Zero();
Element multi_value = 1;
for (int i=0;i< this->_row;i++) multi_value *= this->_vecs[i].data[i];
for (int i=0;i< this->_row;i++){
_mat._vecs[i].data[i] = multi_value / this->_vecs[i].data[i];
}
_mat.spc.digUp = true;
_mat.spc.symmetry = true;
return _mat;
}
else
{
auto vecs = new Vector[this->_row];
for (int row=0;row< this->_row;row++){
auto _data = new Element [this->_column];
for(int col=0;col< this->_column;col++){
Element k = power(-1, row+col);
auto A_ij = getRemainder(col, row).det() * k;
_data[col] = A_ij;
}
auto vec = Vector(_data);
delete[] _data;
vecs[row] = vec;
}
return Matrix(vecs, this->_row, this->_column);
}
}
Matrix Matrix::getAccompanyT() {
if (!isSquare()){ error_models(4); exit(EXIT_FAILURE); }
if (isSingularMat()) { error_models(10);exit(EXIT_FAILURE);}
elif (this->spc.unit) {return copy_mat(*this);}
elif (this->spc.digUp){
auto _mat = Zero();
Element multi_value = 1;
for (int i=0;i< this->_row;i++) multi_value *= this->_vecs[i].data[i];
for (int i=0;i< this->_row;i++){
_mat._vecs[i].data[i] = multi_value / this->_vecs[i].data[i];
}
_mat.spc.digUp = true;
_mat.spc.symmetry = true;
return _mat;
}
else{
auto vecs = new Vector[this->_row];
for (int i=0;i< this->_row;i++){
auto _data = new Element [this->_row];
for (int j=0;j< this->_row;j++){
_data[j] = getRemainder(i, j).det() * power(-1, i+j);
}
vecs[i] = Vector(_data, _column);
delete[] _data;
}
return Matrix(vecs, _row, _row).transpose();
}
}
Matrix Matrix::innerMulti(Matrix & other) {
if (this->_column != other._row) {
error_models(7);
exit(EXIT_FAILURE);
}
if (other.spc.unit) { return copy_mat(*this); }
auto auxiliaryMartrix = other.transpose();
auto vecs = new Vector[this->_row];
for (int i=0;i< this->_row;i++){
auto _data = new Element[other._column];
for (int j=0;j<other._column;j++){
_data[j] = this->_vecs[i].inner(auxiliaryMartrix._vecs[j]);
}
vecs[i] = Vector(_data, other._column);
delete[] _data;
}
return Matrix(vecs, this->_row, other._column);
}
// [优化] 逆矩阵求解
// 保留特殊矩阵（单位阵、主/副对角阵）的 O(1)/O(n) 快速路径
// 一般情况改为高斯-约旦消元法，O(n³) 替代原伴随矩阵法 O(n⁵)
// [BugFix #8] 副对角线矩阵逆矩阵：修复索引错误（原来写入最后一列而非副对角线位置）
Matrix Matrix::inv() {
if (!isSquare()) { error_models(4);exit(EXIT_FAILURE); }
if(isSingularMat()){ error_models(10);exit(EXIT_FAILURE);}
if (this->spc.unit) {return copy_mat(*this);}
elif (this->spc.digUp){
auto _zero = Zero();
for (int i=0;i< this->_row;i++){
_zero._vecs[i].data[i] = 1/ this->_vecs[i].data[i];
}
_zero.spc.digUp = true;
_zero.spc.symmetry = true;
return _zero;
}
elif(this->spc.digDown){
// 副对角线矩阵的逆仍为副对角线矩阵
// A^{-1}[i][n-1-i] = 1 / A[n-1-i][i]
auto _zero = Zero();
int n = this->_row;
for (int i = 0; i < n; i++){
_zero._vecs[i].data[n - 1 - i] = 1.0 / this->_vecs[n - 1 - i].data[i];
}
_zero.spc.digDown = true;
return _zero;
}
else {
// 高斯-约旦消元法求逆，O(n³)
int n = this->_row;
// 增广矩阵 [A | I]，宽度 2n
Element aug[MAXCOUNT][MAXCOUNT * 2];
// 初始化增广矩阵
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
aug[i][j] = this->_vecs[i].data[j];
}
for (int j = 0; j < n; j++) {
aug[i][n + j] = (i == j) ? 1.0 : 0.0;
}
}
// 高斯-约旦消元（带部分列主元）
for (int col = 0; col < n; col++) {
// 选主元
int max_row = col;
Element max_val = fabs(aug[col][col]);
for (int row = col + 1; row < n; row++) {
if (fabs(aug[row][col]) > max_val) {
max_val = fabs(aug[row][col]);
max_row = row;
}
}
// 交换行
if (max_row != col) {
for (int j = 0; j < 2 * n; j++) {
Element temp = aug[col][j];
aug[col][j] = aug[max_row][j];
aug[max_row][j] = temp;
}
}
// 主元归一化
Element pivot = aug[col][col];
for (int j = 0; j < 2 * n; j++) {
aug[col][j] /= pivot;
}
// 消去当前列的其他所有行
for (int row = 0; row < n; row++) {
if (row != col) {
Element factor = aug[row][col];
for (int j = 0; j < 2 * n; j++) {
aug[row][j] -= factor * aug[col][j];
}
}
}
}
// 从增广矩阵右半部分提取逆矩阵
auto vecs = new Vector[n];
for (int i = 0; i < n; i++) {
auto _data = new Element[n];
for (int j = 0; j < n; j++) {
_data[j] = aug[i][n + j];
}
vecs[i] = Vector(_data, n);
delete[] _data;
}
return Matrix(vecs, n, n);
}
}
Element Matrix::trace() const {
// 矩阵的迹就没有必要使用spc特性简化了
if (!isSquare()){ error_models(4); exit(EXIT_FAILURE);}
Element _sum = 0;
for(int i=0;i< this->_row;i++) { _sum += this->_vecs[i].data[i]; }
return _sum;
}
bool Matrix::isOrthogon() {
auto T = this->transpose();
auto E = this->E();
return (*this).innerMulti(T) == E;
}
Matrix Matrix::norm(bool if_line) {
// 行单位化
auto copy = copy_mat(*this);
if(if_line){
for (int i=0;i< this->_row;i++){
copy._vecs[i] = copy._vecs[i].unit();
}
} else{
copy = copy.transpose();
for (int i=0;i< this->_column;i++){
copy._vecs[i] = copy._vecs[i].unit();
}
copy = copy.transpose();
}return copy;
}