高斯整数环学习心得随笔及其实现

Posted 2022-02-01 hans774882968

高斯整数环学习心得随笔

过些天补上~

代码

• ==：两个高斯整数相伴，即为相等。
• 小于和大于：用复数的范数Norm来比较。应用场景举例：高斯整数环的唯一分解定理证明。
• 整除和取模（带余除法）：复数除法的结果是个有理数，但为了方便还是直接用double存了。设两个高斯整数a,b，复数除法结果为a / b = x+yi，则一种可行的取法是round(x)+round(y)*i，这就是整除结果。而g = (x-round(x))+(y-round(y))*i就是带余除法定义中一个可行的“范数小于等于1/2的复数”，g*b即取模结果。具体看代码。
• gcd：既然有带余除法，直接运行辗转相除法即可。

#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
#define rep(i,a,b) for(int i = (a);i <= (b);++i)
#define re_(i,a,b) for(int i = (a);i < (b);++i)
#define dwn(i,a,b) for(int i = (a);i >= (b);--i)

const int N = 1e3 + 5;
const int mod = 998244353;
int dx[4] = -1,1,0,0,dy[4] = 0,0,-1,1;

void dbg()puts("");
template<typename T, typename... R>void dbg(const T &f, const R &... r) 
    cout << f << " ";
    dbg(r...);

template<typename Type>inline void read(Type &xx)
    Type f = 1;char ch;xx = 0;
    for(ch = getchar();ch < '0' || ch > '9';ch = getchar()) if(ch == '-') f = -1;
    for(;ch >= '0' && ch <= '9';ch = getchar()) xx = xx * 10 + ch - '0';
    xx *= f;

void read()
template<typename T,typename ...R>void read(T &x,R &...r)
    read(x);
    read(r...);


struct GI;
LL norm(const GI &x);
GI gcd(const GI &x,const GI &y);

struct GI
    int a,b;
    GI():a(0),b(0)
    GI(int a,int b):a(a),b(b)
    //相伴
    bool operator == (const GI &rh) const
        return a == rh.a && b == rh.b ||
        a == -rh.a && b == -rh.b ||
        a == -rh.b && b == rh.a ||
        a == rh.b && b == -rh.a;
    
    bool operator < (const GI &rh) const
        return norm(*this) < norm(rh);
    
    bool operator <= (const GI &rh) const
        return norm(*this) <= norm(rh);
    
    bool operator > (const GI &rh) const
        return norm(*this) > norm(rh);
    
    bool operator >= (const GI &rh) const
        return norm(*this) >= norm(rh);
    
    GI operator + (const GI &rh) const
        return a+rh.a,b+rh.b;
    
    GI operator - (const GI &rh) const
        return a-rh.a,b-rh.b;
    
    GI operator * (const GI &rh) const
        return a*rh.a-b*rh.b,a*rh.b+b*rh.a;
    
    //整除
    GI operator / (const GI &rh) const
        double x = 1.0*(a*rh.a+b*rh.b)/norm(rh),y = 1.0*(b*rh.a-a*rh.b)/norm(rh);
        return round(x),round(y);
    
    GI operator % (const GI &rh) const
        double x = 1.0*(a*rh.a+b*rh.b)/norm(rh),y = 1.0*(b*rh.a-a*rh.b)/norm(rh);
        complex<double> g = x-round(x),y-round(y);
        g *= complex<double>(rh.a,rh.b);
        return round(g.real()),round(g.imag());
    
    GI operator %= (const GI &rh)
        GI u = (*this) % rh;
        a = u.a;b = u.b;
        return u;
    
    friend ostream& operator << (ostream &sm,const GI &x)
        sm << "(" << x.a << "," << x.b << ")";
        return sm;
    
;
LL norm(const GI &x)return x.a*x.a+x.b*x.b;
GI gcd(const GI &x,const GI &y)
    return y == GI() ? x : gcd(y,x%y);


void test_op()
    dbg("test_op");
    GI x(4,3),y(3,4);
    assert(GI(4,3) == GI(-4,-3) && GI(4,3) == GI(-3,4) && GI(4,3) == GI(3,-4));
    assert(!(x == y) && x <= y && x >= y);
    GI z(4,4);
    assert(x < z && x <= z && z > x && z >= x);
    assert(x+y == GI(7,7));
    assert(x*y == GI(0,25));


//对拍：https://jingyan.baidu.com/article/e75aca8526c6a7142fdac67f.html
void test_div()
    dbg("test_div");
    GI u = GI(1,8) % GI(-1,5);
    dbg(u);//(-2,-3)
    assert(u == GI(2,3));
    
    GI x(18,5),y(7,5);
    u = x - x / y * y;
    dbg(x / y,u);//x / y = (2,-1), u = (-1,2)
    dbg(x % y);//(-1,2)
    assert(u == x % y);
    dbg(y / u);//(1,-4)
    dbg(y % u);//(0,-1)
    
    x = 112,1;
    y = -57,79;
    dbg(x / y);//(-1,-1)
    u = x % y;
    dbg(u);//(-24,23)
    assert(u == GI(-24,23));
    
    x %= y;swap(x,y);
    u = x % y;
    dbg(u);//(-8,-14)
    assert(u == GI(-8,-14));
    
    x %= y;swap(x,y);
    u = x % y;
    dbg(u);//(-4,-7)
    assert(u == GI(4,7));


void test_gcd()
    dbg("test_gcd");
    GI g = gcd(GI(18,5),GI(7,5));
    dbg(g);//(0,-1)
    assert(g == GI(0,-1));
    g = gcd(GI(112,1),GI(-57,79));
    dbg(g);//(-4,-7)
    assert(g == GI(4,7));
    g = gcd(GI(32,9),GI(4,11));
    dbg(g);//(0,-1)
    assert(g == GI(1,0));
    g = gcd(GI(11,3),GI(1,8));
    dbg(g);//(1,-2)
    assert(g == GI(1,-2));
    g = gcd(GI(4,5),GI(4,-5));
    dbg(g);//(0,1)
    assert(g == GI(1,0));


int main(int argc, char** argv) 
    test_op();
    test_div();
    test_gcd();
    return 0;