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几个线代大作业的思路

发布于：2022年11月13日

matrix.h

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#include <cassert>
#include <iostream>
#include <vector>
#include <cstring>
//template<typename Num>
#define Num double
struct Matrix{
    std::string Message;
    std::vector<std::vector<Num> >M;
    int row,col;//行数、列数
    Matrix(int rows,int cols){
        //初始化为零矩阵
//        std::cout<<rows<<" "<<cols;
        row=rows,col=cols;
        M.resize(row+1);
        for (int i=1;i<=row;++i){
            M[i].resize(col+1,0);
        }
        Message="OK";
//        std::cout<<"OK"<<std::endl;
    }
    void init(int rows,int cols){
        row=rows,col=cols;
        M.resize(row+1);
        for (int i=1;i<=row;++i){
            M[i].resize(col+1,0);
        }
        Message="OK";
    }
    Matrix(){
        row=col=0;
    }
    void Input(){
        std::cout<<"Please enter the row and col of the matrix"<<std::endl;
        std::cin>>row>>col;
        assert(row>=1 && col >=1);
        std::cout<<"Please enter the matrix"<<std::endl;
        init(row,col);
        for (int i=1;i<=row;++i){
            for (int j=1;j<=col;++j){
                std::cin>>M[i][j];
            }
        }
    }
    void Output(){
    	std::cout<<"--------------"<<std::endl;
        for (int i=1;i<=row;++i){
            for (int j=1;j<=col;++j){
                std::cout<<M[i][j]<<" ";
            }
            std::cout<<std::endl;
        }
    }
    std::vector<Num>&operator[](int i){
        return M[i];
    }
    void identityMatrix(){
        assert(row==col);
        for (int i=1;i<=row;++i){
            M[i][i]=1;
        }
    }
    void diag(std::vector<Num>V){
        Matrix((int)V.size(),(int)V.size());
        for (int i=0;i<(int)V.size();++i){
            M[i+1][i+1]=V[i];
        }
    }
    //矩阵的初等行、列变换，type='R'行，type='C'列
    void swap(char type,int x,int y){
        if (type=='R'){
            assert(1<=x && x<=row && 1<=y && y<=row);
            for (int i=1;i<=col;++i){
                std::swap(M[x][i],M[y][i]);
            }
        }
        else if (type=='C'){
            assert(1<=x && x<=col && 1<=y && y<=col);
            for (int i=1;i<=row;++i){
                std::swap(M[i][x],M[i][y]);
            }
        }
        else {
            assert(0);
        }
    }
    void times(char type,int x,Num t){
        if (type=='R'){
            assert(1<=x && x<=row);
            for (int i=1;i<=col;++i){
                M[x][i]=t*M[x][i];
            }
        }
        else if (type=='C'){
            assert(1<=x && x<=col);
            for (int i=1;i<=row;++i){
                M[i][x]=t*M[i][x];
            }
        }
        else {
            assert(0);
        }
    }
    void addtimes(char type,int x,Num t,int y){//x行/列的t倍加到y上
        if (type=='R'){
            assert(1<=x && x<=row && 1<=y && y<=row);
            for (int i=1;i<=col;++i){
                M[y][i]=M[y][i]+t*M[x][i];
            }
        }
        else if (type=='C'){
            assert(1<=x && x<=col && 1<=y && y<=col);
            for (int i=1;i<=row;++i){
                M[i][y]=M[i][y]+t*M[i][x];
            }
        }
        else {
            assert(0);
        }
    }
    Num sum(char type,int x){
    	if (type=='R'){
    		Num s=0;
            for (int i=1;i<=col;++i){
                s=s+M[x][i];
            }
            return s;
        }
        else if (type=='C'){
    		Num s=0;
            for (int i=1;i<=row;++i){
                s=s+M[i][x];
            }
            return s;
        }
        else {
            assert(0);
        }
        return 0;
	}
    Matrix transpose(){//矩阵的转置
	    Matrix B(col,row);
	    for (int i=1;i<=B.row;++i){
	        for (int j=1;j<=B.col;++j){
	            B[i][j]=M[j][i];
	        }
	    }
	    return B;
	}
};
Matrix inverse(Matrix A){//求逆矩阵
    assert(A.row==A.col);
    int n=A.row;
    Matrix B(A.row,A.col);
    B.identityMatrix();
    for (int i=1;i<=n;++i){
        int r=i;
        for (int j=i+1;j<=n;++j){
            if (abs(A[j][i])>abs(A[r][i])) r=j;
        }
        A.swap('R',i,r),B.swap('R',i,r);
        if (A[i][i]==0){
            B.Message="Not full rank!";
            return B;
        }
        Num inv=((Num)(1))/A[i][i];
        for (int j=1;j<=n;++j){
            if (j==i) continue;
            Num t=A[j][i]*inv;
            A.addtimes('R',i,t,j);
            B.addtimes('R',i,t,j);
        }
    }
    for (int i=1;i<=n;++i){
        Num inv=((Num)(1))/A[i][i];
        A.times('R',i,inv);
        B.times('R',i,inv);
    }
    B.Message="OK";
    return B;
}
Matrix operator * (Matrix A,Matrix B){
    assert(A.col==B.row);
    Matrix C(A.row,B.col);
    for (int i=1;i<=A.row;++i){
        for (int j=1;j<=B.col;++j){
            for (int k=1;k<=A.col;++k){
                C[i][j]+=A[i][k]*B[k][j];
            }
        }
    }
    return C;
}
Matrix operator * (Num lambda,Matrix A){
    Matrix B(A.row,A.col);
    for (int i=1;i<=A.row;++i){
        for (int j=1;j<=B.col;++j){
            B[i][j]=lambda*A[i][j];
        }
    }
    return B;
}
Matrix operator + (Matrix A,Matrix B){
	assert(A.row==B.row && A.col==B.col);
    Matrix C(A.row,A.col);
    for (int i=1;i<=A.row;++i){
        for (int j=1;j<=A.col;++j){
            C[i][j]=A[i][j]+B[i][j];
        }
    }
    return C;
}
Matrix operator / (Num x,Matrix A){
    return x*inverse(A);
}
Matrix operator / (Matrix A,Matrix B){
    return A*inverse(B);
}
Matrix operator ^ (Matrix A,int k){
	assert(A.row==A.col);
	k--;
	Matrix base=A;
	while (k){
		if (k&1) base=base*A;
		A=A*A;
		k>>=1;
	}
	return base;
}
Num tr(Matrix A){
	assert(A.row==A.col);
	Num ans=0;
	for (int i=1;i<=A.row;++i){
		ans=ans+A[i][i];
	}
	return ans;
}
Matrix addHorizontal(Matrix A,Matrix B){
	assert(A.row==B.row);
	Matrix C(A.row,A.col+B.col);
	for (int i=1;i<=A.row;++i){
		for (int j=1;j<=A.col;++j){
			C[i][j]=A[i][j];
		}
		for (int j=1;j<=B.col;++j){
			C[i][A.col+j]=B[i][j];
		}
	}
	return C;
}
Matrix addVertical(Matrix A,Matrix B){
	assert(A.col==B.col);
	Matrix C(A.row+B.row,A.col);
	for (int i=1;i<=A.col;++i){
		for (int j=1;j<=A.row;++j){
			C[j][i]=A[j][i];
		}
		for (int j=1;j<=B.row;++j){
			C[A.row+j][i]=B[j][i];
		}
	}
	return C;
}

T1

间接胜就是矩阵乘法。如果矩阵是一次操作，代表将原先的队伍（单位矩阵）转换为其胜的队伍，那么两次操作就代表把原先的队伍转换为间接胜的队伍。

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#include "matrix.h"
#include <algorithm>
int main(){
    Matrix A(5,5);
    A[1][2]=1,A[1][3]=1;
    A[2][3]=1,A[2][4]=1,A[2][5]=1;
    A[4][1]=1,A[4][3]=1,A[4][5]=1;
    A[5][1]=1,A[5][3]=1;
    Matrix B=A*A;
    A.Output(),(A*A).Output();
    std::vector<std::pair<int,std::pair<int,int> > >rk;
    for (int i=1;i<=5;++i){
    	rk.push_back(std::make_pair(-A.sum('R',i),std::make_pair(-B.sum('R',i),i)));
	}
	std::sort(rk.begin(),rk.end());
	for (int i=0;i<(int)rk.size();++i){
		std::cout<<rk[i].second.second<<" ";
	}
}

T2

经典的模型，简单的。

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#include "matrix.h"
#include <algorithm>
int main(){
    Matrix A(7,6);
	A.Input();//实在不想手敲了。
	/*
	1 1 0 1 0 0
	1 0 1 1 0 0
	1 1 1 1 0 0
	1 1 0 1 0 0
	1 1 0 1 0 0
	1 0 0 0 1 0
	0 0 0 0 1 1
	*/
	Matrix alpha(6,1);
	alpha[1][1]=1,alpha[4][1]=1,alpha[2][1]=1;
	for (int i=1;i<=7;++i){
		if (alpha.sum('C',1)==(A*alpha).sum('R',i)){
			std::cout<<i<<" matches"<<std::endl;
		}
	}
}

T3,4,5

简单解方程。

T6

见：https://gaisaiyuno.github.io/archives/f2c32ff8.html

T8

用矩阵操作，将 α=(a1,b1,c1),β=(a2,b2,c2)\alpha=(a_1,b_1,c_1),\beta=(a_2,b_2,c_2) 转换为子代的基因型，考虑实二次型 f=xTAxf=x^TAx，这里，将 xTx^T 和 xx 分别改成 αT\alpha^T 和 β\beta 即可。

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#include "matrix.h"
#include <algorithm>
Matrix A1(3,3),A2(3,3),A3(3,3);
Matrix solve(Matrix alpha,Matrix beta){
	Matrix ret(3,1);
	ret[1][1]=(alpha.transpose()*A1*beta)[1][1];
	ret[2][1]=(alpha.transpose()*A2*beta)[1][1];
	ret[3][1]=(alpha.transpose()*A3*beta)[1][1];
	return ret;
}
int main(){
	A1[1][1]=1,A1[2][2]=0.25,A1[1][2]=A1[2][1]=0.5;
	A2[1][2]=A2[2][3]=A2[2][1]=A2[3][2]=0.5,A2[1][3]=A2[3][1]=1,A2[2][2]=0.5;
	A3[3][3]=1,A3[2][2]=0.25,A3[2][3]=A3[3][2]=0.5;
	//Plan 1
	Matrix start(3,1);
	start[1][1]=start[2][1]=start[3][1]=1.0/3.0;
	for (int i=1;i<=10;++i){
		Matrix alpha(3,1);
		alpha[1][1]=1;
		Matrix beta(3,1);
		beta=start;
		start=solve(alpha,beta);
		start.Output();
	}
	//plan 2
	start[1][1]=start[2][1]=start[3][1]=1.0/3.0;
	for (int i=1;i<=10;++i){
		Matrix alpha(3,1);
		alpha[2][1]=1;
		Matrix beta(3,1);
		beta=start;
		start=solve(alpha,beta);
		start.Output();
	}
	//plan 3
	start[1][1]=start[2][1]=start[3][1]=1.0/3.0;
	for (int i=1;i<=10;++i){
		Matrix a(3,1),b(3,1),c(3,1);
		a[1][1]=start[1][1],b[2][1]=start[2][1],c[3][1]=start[3][1];
		start=solve(a,a)+solve(b,b)+solve(c,c);
		start=(1.00/start.sum('C',1))*start;
		start.Output();
	}
}

T10

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#include "matrix.h"
#include <algorithm>
int main(){
    Matrix A(2,2);
    A[1][1]=1,A[1][2]=1,A[2][1]=1;
    Matrix Base(2,2);
	Base.identityMatrix();
	for (int i=1;i<=10;++i){
		Base=Base*A;
		Base.Output();
	} 
}
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本文永久链接是：https://gaisaiyuno.github.io/archives/635c73e3.html

更新于：2022年11月14日

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