趣味程序 - 分数类与矩阵类的实现
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分数和矩阵其实都是Matlab玩剩下的，程序本身纯粹算是C++的一个练习，实现一些基本的功能接口。大体来说就是利用类和重载，最主要的还是体会C++代码复用、封装和标准化接口的思想。C++单论代码效率上当然无法和C相提并论，但是从工程的角度上来说，C++堪称优秀。
唯一的亮点（至少个人看来）在于，行列式的值算法其实很值得研究，如果按照行列式按行/列展开计算则需要递归，复杂度高达
不过现在行列式计算仍然没有非常高效的算法，效率最高的 Fast matrix multiplication
/** MIT license* Copyright © 2016 Maristie* * Permission is hereby granted, free of charge, to any person obtaining a copy of this software* and associated documentation files (the "Software"), to deal in the Software without restriction,* including without limitation the rights to use, copy, modify, merge, publish, distribute, * sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, * subject to the following conditions:* The above copyright notice and this permission notice shall be included in all copies or * substantial portions of the Software.* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED* TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. * IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, * WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE* OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.*/#include<iostream>#include<math.h>#include<time.h>using namespace std;class Fraction;class Matrix;long long findGCD(long long, long long); //find greatest common divisorostream& operator<<(ostream&, Fraction&);istream& operator >> (istream&, Fraction&);ostream& operator<<(ostream&, Matrix&);istream& operator >> (istream&, Matrix&);Matrix operator*(int, Matrix);class Fraction//create Fraction class{private: long long Up, Down; //numerator and denominator int whatSign();//judge sign of fraction bool isInt();//simplize fraction and judge if it's a intpublic: Fraction(); Fraction(int, int); Fraction(const Fraction&);//3 constructors void Simplize();//simplize fraction by divide GCD Fraction operator+(Fraction&); Fraction operator-(Fraction&); Fraction operator*(Fraction&); Fraction operator/(Fraction&);//4 overloaded functions Fraction operator*(int); bool operator==(Fraction); bool operator==(int); bool operator==(long long); bool operator>(Fraction); bool operator<(Fraction); bool operator>=(Fraction); bool operator<=(Fraction); friend ostream& operator<<(ostream&, Fraction&); friend istream& operator >> (istream&, Fraction&); friend class Matrix;};class Matrix//create Matrix class{private: int Row, Col; Fraction** Data; Matrix AlgCof(int, int);//get algebraic cofactor Fraction detAlgCof(int, int);//get the determinant value of algebraic cofactor int maxIndexRow();//get row of the max element int maxIndexCol();//get col of the max elementpublic: Matrix(); Matrix(int, int); Matrix(const Matrix&); ~Matrix(); void setFree();//set Data pointer to free status void Reset(int, int); void Reset(const Matrix&);//reset Data pointer, or rather reallocate space for Data Fraction getElem(int, int); //get element void putElem(int, int, Fraction); //put element in void operator=(Matrix&);//matrix assignment bool operator==(Matrix&); bool operator!=(Matrix&);//compare functions Matrix operator+(Matrix&); Matrix operator-(Matrix&); Matrix operator*(Matrix&); Matrix operator*(int); friend Matrix operator*(int, Matrix); //5 overloaded calculating functions Matrix Power(int); void resize(int, int); void swaprows(int, int); void swapcols(int, int); int size();//return size friend ostream& operator<<(ostream&, Matrix&); friend istream& operator >> (istream&, Matrix&); void transpose();//the result will be output void diag();//here I'll output all elements on the diagonal in order if Row equals Col Fraction det();//a fraction will be returned, and the algorithm complexity is O(n^3) Matrix inv();//a matrix will be returned as the inverse matrix void norm();//the output type is double Fraction max(); void max_index();//output the index in the format (x,y), x=row+1, y=col+1};Fraction::Fraction(){ Up = 0; Down = 1;}Fraction::Fraction(int inputUp, int inputDown){ if (inputDown != 0) { Up = inputUp; Down = inputDown; Simplize(); } else cout << "Denominator can't be zero." << endl;}Fraction::Fraction(const Fraction& origFrac){ Up = origFrac.Up; Down = origFrac.Down;}Fraction Fraction::operator+(Fraction& origFrac){ Fraction Res; Res.Up = Up * origFrac.Down + Down * origFrac.Up; Res.Down = Down * origFrac.Down; Res.Simplize(); return Res;}Fraction Fraction::operator-(Fraction& origFrac){ Fraction Res; Res.Up = Up * origFrac.Down - Down * origFrac.Up; Res.Down = Down * origFrac.Down; Res.Simplize(); return Res;}Fraction Fraction::operator*(Fraction& origFrac){ Fraction Res; Res.Up = Up * origFrac.Up; Res.Down = Down * origFrac.Down; Res.Simplize(); return Res;}Fraction Fraction::operator/(Fraction& origFrac){ if (origFrac == 0) { cout << "0 is divided." << endl; return origFrac; } else { Fraction Res; Res.Up = Up * origFrac.Down; Res.Down = Down * origFrac.Up; Res.Simplize(); return Res; }}Fraction Fraction::operator*(int Num){ Fraction Res(*this); Res.Up *= Num; Res.Simplize(); return Res;}int Fraction::whatSign(){ if (Up * Down == 0) return 0; else if (Up * Down > 0) return 1; else return -1;}void Fraction::Simplize(){ if (whatSign() == 0) { Down = 1; return; } else { int GCD = findGCD(abs(Up), abs(Down)); Up = whatSign() * abs(Up) / GCD; Down = abs(Down) / GCD; return; }}bool Fraction::operator==(Fraction anothFrac){ Fraction myFrac(*this); if (myFrac.Up == anothFrac.Up && myFrac.Down == anothFrac.Down) return true; else return false;}bool Fraction::operator==(int Num){ if (Up == Down * Num) return true; else return false;}bool Fraction::operator==(long long Num){ if (Up == Down * Num) return true; else return false;}bool Fraction::operator>(Fraction origFrac){ Fraction myFrac(*this); if (myFrac.Up * origFrac.Down > origFrac.Up * myFrac.Down) return true; else return false;}bool Fraction::operator>=(Fraction origFrac){ Fraction myFrac(*this); if (myFrac.Up * origFrac.Down >= origFrac.Up * myFrac.Down) return true; else return false;}bool Fraction::operator<(Fraction origFrac){ Fraction myFrac(*this); if (myFrac.Up * origFrac.Down < origFrac.Up * myFrac.Down) return true; else return false;}bool Fraction::operator<=(Fraction origFrac){ Fraction myFrac(*this); if (myFrac.Up * origFrac.Down <= origFrac.Up * myFrac.Down) return true; else return false;}bool Fraction::isInt(){ if (Up == 0 || Down == 1) return true; else return false;}ostream& operator<<(ostream& cout, Fraction& myFrac){ if (myFrac.isInt() == true) cout << myFrac.Up; else cout << myFrac.Up << '/' << myFrac.Down; return cout;}istream& operator >> (istream& cin, Fraction& myFrac){ char Input[1000] = { 0 }; int isFrac = 0; int Pos = 0; cin >> Input; int InputLen = strlen(Input); for (Pos; Pos < InputLen; ++Pos) if (Input[Pos] == '/') { isFrac = 1; break; } if (isFrac == 0) { myFrac.Up = atoi(Input); myFrac.Down = 1; } else { Input[Pos] = 0; char InputUp[1000] = { 0 }; strcpy(InputUp, Input); char InputDown[1000] = { 0 }; strcpy(InputDown, Input + Pos + 1); myFrac.Up = atoi(InputUp); myFrac.Down = atoi(InputDown); } return cin;}long long findGCD(long long Num1, long long Num2) //find GCD by Euclidean Algorithm{ if (Num1 < Num2) { long long SwapTemp = Num1; Num1 = Num2; Num2 = SwapTemp; } long long Remain = Num1 % Num2; while (Remain) { Num1 = Num2; Num2 = Remain; Remain = Num1 % Num2; } return Num2;}Matrix::Matrix(){ Row = Col = 0; Data = NULL;}Matrix::Matrix(int InRow, int InCol){ Reset(InRow, InCol);}Matrix::Matrix(const Matrix& origMat){ Reset(origMat);}void Matrix::Reset(int InRow, int InCol){ Row = InRow; Col = InCol; Data = new Fraction*[Row]; for (int i = 0; i < Row; ++i) Data[i] = new Fraction[Col]; for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) Data[i][j] = Fraction(0, 1);}void Matrix::Reset(const Matrix& origMat){ Row = origMat.Row; Col = origMat.Col; Data = new Fraction*[Row]; for (int i = 0; i < Row; ++i) { Data[i] = new Fraction[Col]; for (int j = 0; j < Col; ++j) Data[i][j] = origMat.Data[i][j]; }}Matrix::~Matrix(){ setFree();}void Matrix::setFree(){ for (int i = 0; i < Row; ++i) delete[] Data[i]; delete[] Data; Data = NULL;}Fraction Matrix::getElem(int i, int j){ return Data[i][j];}void Matrix::putElem(int i, int j, Fraction origFrac){ Data[i][j] = origFrac; return;}void Matrix::operator=(Matrix& origMat){ setFree(); Reset(origMat.Row, origMat.Col); for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) Data[i][j] = origMat.Data[i][j]; return;}bool Matrix::operator==(Matrix& origMat){ if (Row == origMat.Row && Col == origMat.Col) { for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) if (!(Data[i][j] == origMat.Data[i][j])) return false; return false; } else cout << "Illegal input." << endl; return true;}bool Matrix::operator!=(Matrix& origMat){ if (Row == origMat.Row && Col == origMat.Col) { for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) if (!(Data[i][j] == origMat.Data[i][j])) return true; return false; } else cout << "Illegal input." << endl; return false;}Matrix Matrix::operator+(Matrix& origMat){ if (Row == origMat.Row && Col == origMat.Col) { Matrix resMat(Row, Col); for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) resMat.Data[i][j] = Data[i][j] + origMat.Data[i][j]; return resMat; } else cout << "Illegal input." << endl; return origMat;}Matrix Matrix::operator-(Matrix& origMat){ if (Row == origMat.Row && Col == origMat.Col) { Matrix resMat(Row, Col); for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) resMat.Data[i][j] = Data[i][j] - origMat.Data[i][j]; return resMat; } else cout << "Illegal input." << endl; return origMat;}Matrix Matrix::operator*(Matrix& origMat){ if (Col == origMat.Row) { Matrix resMat(Row, origMat.Col); for (int i = 0; i < resMat.Row; ++i) for (int j = 0; j < resMat.Col; ++j) for (int k = 0; k < Col; ++k) resMat.Data[i][j] = resMat.Data[i][j] + Data[i][k] * origMat.Data[k][j]; return resMat; } else cout << "Illegal input." << endl; return origMat;}Matrix Matrix::operator*(int Num){ Matrix resMat(Row, Col); for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) resMat.Data[i][j] = Data[i][j] * Num; return resMat;}Matrix operator*(int Num, Matrix origMat){ Matrix resMat(origMat.Row, origMat.Col); for (int i = 0; i < origMat.Row; ++i) for (int j = 0; j < origMat.Col; ++j) resMat.Data[i][j] = origMat.Data[i][j] * Num; return resMat;}Matrix Matrix::Power(int n){ if (Row != Col || n <= 0) { cout << "Illegal input." << endl; return *this; } else { Matrix resMat(*this); for (int i = 1; i < n; ++i) resMat = resMat * (*this); return resMat; }}void Matrix::resize(int sizeRow, int sizeCol){ if (Row * Col != sizeRow * sizeCol) cout << "Illegal input." << endl; else { Fraction* MemStor = new Fraction[Row * Col]; for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) MemStor[i * Col + j] = Data[i][j]; setFree(); Reset(sizeRow, sizeCol); for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) Data[i][j] = MemStor[i * Col + j]; delete[] MemStor; MemStor = NULL; } return;}void Matrix::swaprows(int Row1, int Row2){ Fraction swapTemp; for (int i = 0; i < Col; ++i) { swapTemp = Data[Row1][i]; Data[Row1][i] = Data[Row2][i]; Data[Row2][i] = swapTemp; } return;}void Matrix::swapcols(int Col1, int Col2){ Fraction swapTemp; for (int j = 0; j < Row; ++j) { swapTemp = Data[j][Col1]; Data[j][Col1] = Data[j][Col2]; Data[j][Col2] = swapTemp; } return;}int Matrix::size(){ return Row * Col;}istream& operator >> (istream& cin, Matrix& myMat){ for (int i = 0; i < myMat.Row; ++i) for (int j = 0; j < myMat.Col; ++j) cin >> myMat.Data[i][j]; return cin;}ostream& operator<<(ostream& cout, Matrix& myMat){ for (int i = 0; i < myMat.Row; ++i) { for (int j = 0; j < myMat.Col; ++j) cout << myMat.Data[i][j] << '\t'; cout << endl; } return cout;}void Matrix::transpose(){ Fraction* memStor = new Fraction[Row * Col]; for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) memStor[i * Col + j] = Data[i][j]; setFree(); Reset(Col, Row); for (int i = 0; i < Col; ++i) for (int j = 0; j < Row; ++j) Data[j][i] = memStor[i * Row + j]; delete[] memStor; memStor = NULL; cout << "After transpose, the matrix turns to:" << endl; cout << *this << endl; return;}void Matrix::diag(){ if (Row != Col) { cout << "No diag element existing." << endl; return; } int i; cout << "Output the diag elements in order:" << endl; for (i = 0; i<Row - 1; ++i) cout << getElem(i, i) << ' '; cout << getElem(i, i) << endl; return;}Fraction Matrix::det(){ Matrix myMat(*this); if (myMat.Row != myMat.Col) { cout << "Row not equal to Col." << endl; return myMat.Data[0][0]; } else { int i, j, k; int SwapTime = 0; for (i = 0; i < myMat.Row; ++i) { if (myMat.Data[i][i] == 0) { int isDetZero = 1; for (j = i + 1; j < myMat.Row; ++j) if (!(myMat.Data[j][i] == 0)) { myMat.swaprows(i, j); ++SwapTime; isDetZero = 0; break; } if (isDetZero == 1) { Fraction resFrac(0, 1); return resFrac; } } for (j = i + 1; j < myMat.Row; ++j) for (k = myMat.Col - 1; k >= i; --k) myMat.Data[j][k] = myMat.Data[j][k] - myMat.Data[i][k] * myMat.Data[j][i] / myMat.Data[i][i]; } Fraction resFrac(1, 1); for (i = 0; i < Row; ++i) resFrac = resFrac * myMat.Data[i][i]; if (SwapTime % 2 == 1) resFrac = resFrac * (-1); return resFrac; }}Matrix Matrix::AlgCof(int InRow, int InCol){ Matrix resMat(Row - 1, Col - 1); int resRow = 0, resCol = 0; for (int i = 0; i < Row; ++i) { if (i == InRow) continue; for (int j = 0; j < Col; ++j) { if (j == InCol) continue; resMat.Data[resRow][resCol] = Data[i][j]; ++resCol; } ++resRow; resCol = 0; } return resMat;}Fraction Matrix::detAlgCof(int InRow, int InCol){ Matrix AlgMat = AlgCof(InRow, InCol); Fraction detValue = AlgMat.det(); if ((InRow + InCol) % 2 == 1) detValue = detValue * -1; return detValue;}Matrix Matrix::inv(){ if (Row != Col) { cout << "Row unequal to Col." << endl; return *this; } else { Matrix resMat(Row, Col); Fraction detVal = det(); if (detVal == 0) { cout << "No inverse matrix." << endl; return *this; } for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) resMat.Data[i][j] = detAlgCof(j, i) / detVal; return resMat; }}void Matrix::norm(){ Fraction Sum; for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) Sum = Sum + Data[i][j] * Data[i][j]; cout << "The norm value is:" << endl; cout << sqrt(1.0*Sum.Up / Sum.Down) << endl << endl; return;}Fraction Matrix::max(){ Fraction Res = Data[0][0]; for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) if (Data[i][j] > Res) Res = Data[i][j]; return Res;}int Matrix::maxIndexRow(){ int Val = 0; Fraction Res = Data[0][0]; for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) if (Data[i][j] > Res) { Res = Data[i][j]; Val = i; } return Val;}int Matrix::maxIndexCol(){ int Val = 0; Fraction Res = Data[0][0]; for (int i = 0; i < Row; ++i) for (int j = 0; j < Col; ++j) if (Data[i][j] > Res) { Res = Data[i][j]; Val = j; } return Val;}void Matrix::max_index(){ int row = maxIndexRow(); int col = maxIndexCol(); cout << "The max index is: (" << row + 1 << ',' << col + 1 << ")." << endl; return;}int main(){ Matrix a(4, 4); srand((unsigned int)time(0)); for (int i = 0; i < 4; ++i) for (int j = 0; j < 4; ++j) a.putElem(i, j, Fraction(rand() % 100 + 10 * i + j + 1, rand() % 100 + 10 * j + i + 1)); cout << a << endl << a.inv() << endl << a.det() << endl << endl; a.transpose(); a.norm(); cout << a.inv().inv() << endl; a.transpose(); return 0;}参考资料：
[1] - Computational complexity of mathematical operations, Wikipedia, https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations
[2] - Strassen algorithm, Wikipedia, https://en.wikipedia.org/wiki/Strassen_algorithm
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