# 高斯消去法
---
# CODE
```cpp
#pragma GCC optimize("Ofast,unroll-loops,no-stack-protector,fast-math")
#include <bits/stdc++.h>
#define IO ios::sync_with_stdio(0),cin.tie(0)
#define F first
#define S second
#define MP make_pair
#define PB push_back
#define PPB pop_back
#define ALL(v) v.begin(),v.end()
#define F0R(i,n) for(int i=0;i<=n;++i)
#define FOR(i,m,n) for(int i=m;i<=n;++i)
#define RFOR(i,m,n) for(int i=m;i>=n;--i)
#define RF0R(i,n) for(int i=n;i>=0;--i)
#define SZ(v) (int) v.size()
#define maxE(v) (*max_element((x).begin(), (x).end()))
#define minE(v) (*min_element((x).begin(), (x).end()))
#define SETW(x) setiosflags(ios::left) << setw(x)
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int, int> PII;
typedef pair<double,double> PDD;
typedef vector<int> VI;
typedef vector<PII> VPII;
typedef vector<LL> VLL;
typedef vector<double> VD;
typedef vector<PDD> VPDD;
typedef __int128 I128;
//--------------------------------------------------------
#define gcd __gcd
constexpr int N=105;
int n;
struct FRAC{int a,b;} mat[N][N];
inline FRAC norm(const FRAC &x){
int aa=x.a,bb=x.b;
int gcd_=gcd(aa,bb);
aa/=gcd_;
bb/=gcd_;
if(bb<0) aa=-aa,bb=-bb;
return {aa,bb};
}
FRAC operator+(const FRAC &x,const FRAC &y){
int gcd_=gcd(x.b,y.b);
int bb=x.b*y.b/gcd_;
int aa=x.a*(y.b/gcd_)+y.a*(x.b/gcd_);
return norm({aa,bb});
}
FRAC operator-(const FRAC &x,const FRAC &y){
int gcd_=gcd(x.b,y.b);
int bb=x.b*y.b/gcd_;
int aa=x.a*(y.b/gcd_)-y.a*(x.b/gcd_);
return norm({aa,bb});
}
FRAC operator*(const FRAC &x,const FRAC &y){
int aa=x.a*y.a,bb=x.b*y.b;
return norm({aa,bb});
}
FRAC operator/(const FRAC &x,const FRAC &y){
FRAC y2={y.b,y.a};
return x*y2;
}
void eli(const int& from,const int& to){
FRAC mul_=mat[to][from]/mat[from][from];
FOR(i,from,n) mat[to][i]=mat[to][i]-(mul_*mat[from][i]);
return;
}
void solve(){
F0R(nc,n-1){
if(mat[nc][nc].a==0)
FOR(j,nc+1,n-1){
if(mat[j][nc].a!=0){
F0R(i,n) swap(mat[nc][i],mat[j][i]);
break;
}
}
if(mat[nc][nc].a==0) continue;
for(int i=nc+1;i<n;i++) eli(nc,i);
}
return;
}
void print(const FRAC &x){
string s=to_string(x.a);
if(x.b!=1) {s+="/";s+=to_string(x.b);}
cout << SETW(10) << s;
return;
}
signed main(){
cout << "請問此方陣之列數為何(1~100)\n";
cin >> n;
cout << "請輸入其增廣矩陣，兩數字間以空格隔開\n";
F0R(i,n-1)F0R(j,n){
mat[i][j].b=1;
cin >> mat[i][j].a;
}
solve();
cout << "---------------------------------------------\n";
cout << "此為進行完高斯消去法後的增廣矩陣:\n";
F0R(i,n-1){
F0R(j,n){
print(mat[i][j]);
cout << ' ';
}
cout << '\n';
}
cout << "---------------------------------------------\n";
bool infinite=false;
F0R(i,n-1){
bool zero=true;
F0R(j,n-1){
if(mat[i][j].a!=0){
zero=false;
break;
}
}
if(zero){
if(mat[i][n].a==0) infinite=true;
else{
cout << "此方程無解\n";
return 0;
}
}
}
if(infinite){
cout << "此方程有無限多組解\n";
return 0;
}
vector<FRAC> ans(n);
RF0R(i,n-1){
FRAC now=mat[i][n];
FOR(j,i+1,n-1){
now=now-(mat[i][j]*ans[j]);
}
ans[i]=now/mat[i][i];
}
cout << "此為未知數之數值:\n";
F0R(i,n-1){
cout << 'a' << i+1 << "= ";
print(ans[i]);
cout << '\n';
}
return 0;
}
```
---
# 整數
---
# 分數
---
# 無解
---
# 無限多組解
{"title":"高斯消去法","contributors":"[{\"id\":\"b3c5fd10-313b-43e1-9849-80b434b3e8ea\",\"add\":11752,\"del\":7753}]","description":"螢幕擷取畫面 2025-02-09 161527"}