高斯消元 模板

照着czyuan的那个模板，手敲了一遍，存一下。

貌似今天一整天就看了一下高斯消元的知识，然后看了模板，又手敲了一遍。

#include <iostream>
#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <cmath>
#include <algorithm>
#define LL __int64
const int maxn = 100+10;
using namespace std;
int equ, var, fn;
int a[maxn][maxn], x[maxn];
bool free_x[maxn];

int gcd(int a, int b)
{
    return b==0?a:gcd(b, a%b);
}
int lcm(int a, int b)
{
    return a*b/gcd(a, b);
}
int Gauss()
{
    int i, j, k, max_r, col;
    int ta, tb, LCM, tmp, fx_num;
    int free_index;
    col = 0;

    for(k = 0; k<equ && col<var; k++, col++)
    {
        max_r = k;
        for(i = k+1; i < equ; i++)
            if(abs(a[i][col])>abs(a[max_r][col]))
            max_r = i;

        if(max_r != k)
            for(j = k; j < var+1; j++)
            swap(a[k][j], a[max_r][j]);

        if(a[k][col]==0)
        {
            k--;
            continue;
        }
        for(i = k+1; i < equ; i++)
        {
            if(a[i][col] != 0)
            {
                LCM = lcm(abs(a[i][col]), abs(a[k][col]));
                ta = LCM/abs(a[i][col]);
                tb= LCM/abs(a[k][col]);
                if(a[i][col]*a[k][col] < 0) tb = -tb;

                for(j = col; j < var+1; j++)
                a[i][j] = a[i][j]*ta - a[k][j]*tb;
            }
        }
    }
    for(i = k; i < equ; i++)
    if(a[i][col] != 0)
    return -1;

    if(k < var)
    {
        for(i = k-1; i >= 0; i--)
        {
            fx_num = 0;
            for(j = 0; j < var; j++)
                if(a[i][j]!=0 && free_x[j])
                {
                    fx_num ++;
                    free_index = j;
                }
            if(fx_num > 1) continue;

            tmp = a[i][var];
            for(j = 0; j < var; j++)
                if(a[i][j] != 0 && j!=free_index)
                tmp -= a[i][j]*x[j];

            x[free_index] = tmp/a[i][free_index];
            free_x[free_index] = 0;
        }
        return var-k;
    }
    for(i = var-1; i >= 0; i--)
    {
        tmp = a[i][var];
        for(j = i+1; j < var; j++)
        if(a[i][j] != 0)
        tmp -= a[i][j]*x[j];

        if(tmp%a[i][i] != 0) return -2;
        x[i] = tmp/a[i][i];
    }
    return 0;
}
int main()
{
    int i, j;
    while(cin>>equ>>var)
    {
        memset(a, 0, sizeof(a));
        memset(x, 0, sizeof(x));
        memset(free_x, 1, sizeof(free_x));
        for(i = 0; i < equ; i++)
        {
            for(j = 0; j < var+1; j++)
            cin>>a[i][j];
        }

        fn = Gauss();
        if(fn == -1) cout<<"无解"<<endl;
        else if(fn == -2)
        cout<<"有浮点数解， 无整数解"<<endl;
        else if(fn > 0)
        {
            printf("无穷多解! 自由变元个数为%d\n", fn);
            for(i = 0; i < var; i++)
            {
                if(free_x[i]) printf("x%d行 是不确定的\n", i + 1);
                else printf("x%d: %d\n", i + 1, x[i]);
            }
        }
        else
        {
            for(i = 0; i < var; i++)
            printf("x%d: %d\n", i + 1, x[i]);
        }
        cout<<endl;
    }
    return 0;
}

 

 

浮点型的模板

eps那里和整型不一样

#define eps 1e-8
const int maxn = 100+10;
using namespace std;
int equ, var;
double a[maxn][maxn], x[maxn];

int Gauss()
{
    int i, j, k, max_r, col;
    double tmp;
    col = 0;

    for(k = 0; k<equ && col<var; k++, col++)
    {
        max_r = k;
        for(i = k+1; i < equ; i++)
            if(fabs(a[i][col])-fabs(a[max_r][col]) > eps)
            max_r = i;

        if(max_r != k)
            for(j = k; j < var+1; j++)
            swap(a[k][j], a[max_r][j]);

        if(fabs(a[k][col]) < eps)
        {
            k--;
            continue;
        }
        for(i = k+1; i < equ; i++)
        {
            if(fabs(a[i][col]) > eps)
            {
                double t = a[i][col]/a[k][col];  //这里和整型的不同。
                a[i][col] = 0.0;

                for(j = col; j < var+1; j++)
                a[i][j] -= a[k][j]*t;
            }
        }
    }
    for(i = var-1; i >= 0; i--)
    {
        if(fabs(a[i][i]) < eps) continue;
        tmp = a[i][var];
        for(j = i+1; j < var; j++)
        if(a[i][j] != 0)
        tmp -= a[i][j]*x[j];

        //if(tmp%a[i][i] != 0) return -2;
        x[i] = tmp/a[i][i];
    }
    return 0;
}

 

 

 

模mo的情况下的模板：

根据上面的模板改的，但是正确性有待验证

 

int gcd(int a, int b)
{
    return b==0?a:gcd(b, a%b);
}
int lcm(int a, int b)
{
    return a*b/gcd(a, b);
}
int Gauss()
{
    int x_mo;
    x_mo = 2;  //初始化为2了。
    int i, j, k, max_r, col;
    int ta, tb, LCM, tmp, fx_num;
    int free_index;
    col = 0;

    for(k = 0; k<equ && col<var; k++, col++)
    {
        max_r = k;
        for(i = k+1; i < equ; i++)
            if(abs(a[i][col])>abs(a[max_r][col]))
                max_r = i;

        if(max_r != k)
            for(j = k; j < var+1; j++)
                swap(a[k][j], a[max_r][j]);

        if(a[k][col]==0)
        {
            k--;
            continue;
        }
        for(i = k+1; i < equ; i++)
        {
            if(a[i][col] != 0)
            {
                LCM = lcm(abs(a[i][col]), abs(a[k][col]));
                ta = LCM/abs(a[i][col]);
                tb= LCM/abs(a[k][col]);
                if(a[i][col]*a[k][col] < 0) tb = -tb;

                for(j = col; j < var+1; j++)
                    a[i][j] = ((a[i][j]*ta - a[k][j]*tb)%x_mo+x_mo)%x_mo;
            }
        }
    }
    for(i = k; i < equ; i++)
        if(a[i][col] != 0)
            return -1;

    if(k < var)  //正确性有待验证
    {
        for(i = k-1; i >= 0; i--)
        {
            fx_num = 0;
            for(j = 0; j < var; j++)
                if(a[i][j]!=0 && free_x[j])
                {
                    fx_num ++;
                    free_index = j;
                }
            if(fx_num > 1) continue;

            tmp = a[i][var];
            for(j = 0; j < var; j++)
                if(a[i][j] != 0 && j!=free_index)
                    tmp = ((tmp-a[i][j]*x[j])%x_mo+x_mo)%x_mo;
            if(a[i][free_index]==0)  //这里我也改了，因为也会出现0的情况。
                x[free_index] = 0;
            else
            {
                while(tmp%a[i][free_index]!=0)  //正确性有待验证
                    tmp += x_mo;  //正确性有待验证
                x[free_index] = (tmp/a[i][free_index])%x_mo; //正确性有待验证
            }
            free_x[free_index] = 0;
        }
        return var-k;
    }

    for(i = var-1; i >= 0; i--)
    {
        tmp = a[i][var];
        for(j = i+1; j < var; j++)
            if(a[i][j] != 0)
                tmp = ((tmp-a[i][j]*x[j])%x_mo+x_mo)%x_mo;

        if(a[i][i]==0) //这我改了，因为做题的时候发现有a[][]等于0的情况，会出现除0
            x[i] = 0;
        else
        {
            if(tmp%a[i][i] != 0) return -2;  //这个的位置应该放这。
            while(tmp%a[i][i]!=0) tmp += x_mo;
            x[i] = (tmp/a[i][i])%x_mo;
        }
    }
    return 0;
}

 

 

 

 

 

这个和上面的模板不一样，也有判断多解和无解的情况。

int Gauss ()
{
    int x_mo;
    x_mo = 7;  //初始为7了。
    int i, j, mr, row, col;
    int ta, tb, l, tmp;
    row = col = 0;
    while ( row < equ && col < var )
    {
        mr = row;
        for ( i = row + 1; i < equ; i++ )
            if ( abs(a[i][col]) > abs(a[mr][col]) ) mr = i;


        if ( mr != row )
            for ( j = col; j <= var; j++ )
                swap ( a[row][j], a[mr][j] );
        if ( a[row][col] == 0 )
        {
            col++;
            continue;
        }


        for ( i = row + 1; i < equ; i++ )
        {
            if ( a[i][col] != 0 )
            {
                l = lcm(abs(a[i][col]), abs(a[row][col]));
                ta = l / abs(a[i][col]), tb = l / abs(a[row][col]);
                if ( a[i][col] * a[row][col] < 0 ) tb = -tb;
                for ( j = col; j <= var; j++ )
                    a[i][j] = ((a[i][j]*ta-a[row][j]*tb)%x_mo+x_mo) % x_mo;
            }
        }
        row++;
        col++;
    }

    for ( i = row; i < equ; i++ )
        if ( a[i][var] != 0 ) return -1;


    if ( row < var )
        return var - row;


    for ( i = var - 1; i >= 0; i-- )
    {
        tmp = a[i][var];
        for ( j = i + 1; j < var; j++ )
            tmp = ((tmp-a[i][j]*x[j])%x_mo+x_mo)%x_mo;
        while ( tmp % a[i][i] != 0 ) tmp += x_mo;
        x[i] = ( tmp / a[i][i] ) % x_mo;
    }
    return 0;
}

 

这个是判断模2 情况下的模板，注意只是模2，不会出现除0的情况，但是没有找自由变元的处理：

int  Gauss()
{
    int i,j,k;
    int max_r;
    int col;
    int temp;

    int free_x_num;
    int free_index;

    col=0;
    for(k=0;k<equ&&col<var;k++,col++)
    {
        max_r=k;
        for(i=k+1;i<equ;i++)
        {
            if(abs(a[i][col])>abs(a[max_r][col]))max_r=i;
        }
        if(max_r!=k)
        {
            for(j=col;j<var+1;j++)swap(a[k][j],a[max_r][j]);
        }
        if(a[k][col]==0)
        {
            k--;
            continue;
        }
        for(i=k+1;i<equ;i++)
        {
            if(a[i][col]!=0)
            {
                for(j=col;j<var+1;j++)
                  a[i][j]^=a[k][j];
            }
        }
    }
    for(i=k;i<equ;i++)
    {
        if(a[i][col]!=0)return -1;//无解
    }

    if(k < var)
    return var-k;

    for(i=var-1;i>=0;i--)
    {
        x[i]=a[i][var];
        for(j=i+1;j<var;j++)
          x[i]^=(a[i][j]&&x[j]);
    }
    return 0;
}