高斯消元整数消元模板

Posted 2022-04-16 ITAK

高斯消元就是来接方程组的。（可以跟矩阵联系在一起）

#include <iostream>
#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <cmath>
using namespace std;
const int MAXN = 1e2+5;
int equ, var;///equ个方程 var个变量
int a[MAXN][MAXN];///增广矩阵
int x[MAXN];///解的数目
bool free_x[MAXN];///判断是不是自由变元
int free_num;///自由变元的个数
inline int GCD(int m, int n)

    if(n == 0)
        return m;
    return GCD(n, m%n);

inline int LCM(int a, int b)

    return a/GCD(a,b)*b;


int Gauss()

    int Max_r;///当前列绝对值最大的存在的行
    ///col：处理当前的列
    int row,col = 0;
    int free_x_num;
    int free_index;
    for(int i=0; i<=var; i++)
    
        x[i] = 0;
        free_x[i] = 1;
    
    for(row=0; row<equ&&col<var; row++,col++)
    
        Max_r = row;
        for(int i=row+1; i<equ; i++)
            if(abs(a[i][col]) > abs(a[Max_r][col]))
                Max_r = i;

        if(Max_r != row)
            for(int i=0; i<var+1; i++)
                swap(a[row][i], a[Max_r][i]);

        if(a[row][col] == 0)
        
            row--;
            continue;
        
        for(int i=row+1; i<equ; i++)
        
            if(a[i][col])
            
                int lcm = LCM(abs(a[i][col]), abs(a[row][col]));
                int tp1=lcm/abs(a[i][col]), tp2=lcm/abs(a[row][col]);
                if(a[row][col]*a[i][col] < 0)
                    tp2 = -tp2;
                for(int j=col; j<var+1; j++)
                    a[i][j] = tp1*a[i][j]-tp2*a[row][j];
            
        
    
    for(int i=row; i<equ; i++)
        if(a[i][col])
            return -1;///无解

    if(row < var)
    
        for(int i=row-1; i>=0; i--)
        
            free_x_num = 0;
            for(int j=0; j<var; j++)
                if(a[i][j] && free_x[j])
                
                    free_x_num++;
                    free_index = j;
                

            if(free_x_num > 1)
                continue;
            int tmp = a[i][var];
            for(int j=0; j<var; j++)
                if(a[i][j] && j!=free_index)
                    tmp -= a[i][j]*x[j];
            x[free_index] = tmp/a[i][free_index];/// 求出该变元.
            free_x[free_index] = 0; /// 该变元是确定的.
        
        return var - row;///自由变元的个数
    
    for(int i=var-1; i>=0; i--)
    
        int tmp = a[i][var];
        for(int j=i+1; j<var; j++)
            if (a[i][j])
                tmp -= a[i][j]*x[j];
        if (tmp%a[i][i])
            return -2; /// 说明有浮点数解，但无整数解.
        x[i] = tmp/a[i][i];
    
    return 0;///唯一解

void Debug()

    puts("");
    cout<<"+++++++++++++++++++++++++++分界线++++++++++++++++++++++++++++++"<<endl;
    for(int i=0; i<equ; i++)
    
        for(int j=0; j<var+1; j++)
        
            cout<<a[i][j]<<" ";
        
        cout<<endl;
    
    cout<<"+++++++++++++++++++++++++++分界线++++++++++++++++++++++++++++++"<<endl;
    puts("");

int main()

    while(cin>>equ>>var)
    
        for(int i=0; i<equ; i++)
        
            for(int j=0; j<var+1; j++)
                cin>>a[i][j];
        
        ///Debug();
        cout<<Gauss()<<endl;
    
    return 0;

/**
4 4
1 2 3 4 3
0 1 2 3 3
0 0 0 1 2
0 0 0 0 0
*/