数学上，高斯消元法（或译：高斯消去法），是线性代数中的一个算法，可用来为线性方程组求解，求出矩阵的秩，以及求出可逆方阵的逆矩阵。当用于一个矩阵时，高斯消元法会产生出一个“行梯阵式”。高斯消元法可以用在电脑中来解决数千条等式及未知数。不过，如果有过百万条等式时，这个算法会十分费时。一些极大的方程组通常会用迭代法来解决。亦有一些方法特地用来解决一些有特别排列的系数的方程组。

 高斯消元法可用来找出下列方程组的解或其解的限制：

---

#include <iostream>
#include <stdio.h>
#include <math.h>
#include <iomanip>
using namespace std;

#define N 3

static double a[N][N]= {{2,6,3},{4,7,7},{-2,4,5}};
static double aa[N][N]= {{2,6,3},{4,7,7},{-2,4,5}};
static double b[N]= {3,1,-7};
static double bb[N]= {3,1,-7};
void print()
{
    for(int i=0; i<N; i++)
    {
        for(int j=0; j<N; j++)
        {
            cout<<setw(15)<<left<<a[i][j];
        }
        cout<<b[i]<<endl;
    }
}
bool check(double *x)
{
    int flag=1;
    for(int i=0;i<N;i++)
    {
        double temp=0;
        for(int j=0;j<N;j++)
        {
            temp=temp+aa[i][j]*x[j];
        }
        if(temp!=bb[i])
        {
            cout<<"方程组解有误！第"<<i+1<<"方程验证不通过"<<endl;
            cout<<"左边="<<showpoint<<temp<<"   右边="<<showpoint<<bb[i]<<endl;
            flag=0;
            //break;
        }
    }
    return flag;
}

int main()
{
    double x[N]= {0};
    cout<<"高斯消去法仅仅可以对系数矩阵为满秩矩阵的方程组求解"<<endl;
    print();
    for(int i=0; i<N-1; i++) ///////////////对每一列找主元消去（最后一列除外）
    {
        double temp=a[i][i];
        int p;
        for(int j=i; j<N; j++) /////////找主元
        {
            if(fabs(a[j][i])>temp)
            {
                temp=fabs(a[j][i]);
                //cout<<temp<<endl;
                p=j;
            }
        }
        //////////////////////////////交换行
        for(int j=i; j<N; j++)
        {
            double tem;
            tem=a[i][j];
            a[i][j]=a[p][j];
            a[p][j]=tem;
        }

        {
            double tem;
            tem=b[p];
            b[p]=b[i];
            b[i]=tem;
        }
        cout<<"找第"<<i+1<<"个主元后："<<endl;
        print();
        /////////////////////////////////////////////
        if(a[i][i]==0)
        {
            cout<<"初等变换后系数矩阵第"<<i<<"列全为0，方程组有无数个解"<<endl;
            break;
        }
        else
        {
            for(int p=i+1; p<N; p++)
            {
                double rand;
                rand=a[p][i]/a[i][i];
                if(rand!=0)
                {
                    cout<<"对第"<<i+1<<"个主元对第"<<p+1<<"行消去(消去系数="<<rand<<")："<<endl;
                    for(int q=i; q<N; q++)
                    {
                        //cout<<"前a["<<p+1<<"]["<<q+1<<"]="<<a[p][q]<<endl;
                        //cout<<"a[i][q]*rand="<<a[i][q]*rand<<endl;
                        a[p][q]=a[p][q]-a[i][q]*rand;
                        //cout<<"后a["<<p+1<<"]["<<q+1<<"]="<<a[p][q]<<endl;
                    }
                    b[p]=b[p]-b[i]*rand;
                    print();
                }
            }
        }
    }
    for(int i=N-1;i>=0;i--)
    {
        double temp=0;
        for(int j=i;j<N;j++)
        {
            if(j>i)
            temp=temp+a[i][j]*x[j];
        }
        x[i]=(b[i]-temp)/a[i][i];

    }
    cout<<"/////////////////////////////////////////////////"<<endl;
    for(int i=0;i<N;i++)
    {
        cout<<"x"<<i+1<<"="<<x[i]<<setw(10)<<right;
    }cout<<endl;
    if(check(x))cout<<"验证成功"<<endl;
    //cout << "Hello world!" << endl;
    return 0;
}