AB fft

A*B fft

P1919 【模板】A*B Problem升级版（FFT快速傅里叶） - 洛谷 | 计算机科学教育新生态 (luogu.com.cn)

參考代碼

#include<iostream>
#include<algorithm>
#include<vector>
#include<cmath>
#include<cstdio>
#include<cstring>
using namespace std;
const int MAX_FFT = 2e7 + 10;
const int MAX_AB = 2e7 + 10;//系数项的两倍

inline int read()
{
    char c = getchar(); int x = 0, f = 1;
    while (c < '0' || c>'9') { if (c == '-')f = -1; c = getchar(); }
    while (c >= '0' && c <= '9') { x = x * 10 + c - '0'; c = getchar(); }
    return x * f;
}
struct complex {//负数的加减乘
    double x, y;
    complex(double _x, double _y) { x = _x; y = _y; }
    complex() {}
    complex operator+(const complex& t) { return { x + t.x,y + t.y }; }
    complex operator-(const complex& t) { return { x - t.x,y - t.y }; }
    complex operator*(const complex& t) { return { x * t.x - y * t.y,x * t.y + y * t.x }; }
};
complex a[MAX_AB], b[MAX_AB];
int n, m;
string s1, s2;
struct  FFT//快速傅里叶变换
{
    //存储
    double Pi = acos(-1.0);
    int l = 0, rev[MAX_FFT];
    int limit = 1;
    void init(int len)//len:为两个多项式乘积最高次幂为(degree(A)+degree(B))
    {
        while (limit <= len) limit <<= 1, l++;
        for (int i = 0; i < limit; i++)
            rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << (l - 1));
    }
    //后面的type表示要进行的变换是什么类型
    //1表示从系数变为点值
    //-1表示从点值变为系数 
    void fast_fast_tle(complex* A, int type)
    {


        for (int i = 0; i < limit; i++)
            if (i < rev[i]) swap(A[i], A[rev[i]]);//求出要迭代的序列 

        for (int mid = 1; mid < limit; mid <<= 1)//待合并区间的中点
        {
            complex Wn(cos(Pi / mid), type * sin(Pi / mid)); //单位根 
            for (int R = mid << 1, j = 0; j < limit; j += R)//R是区间的右端点，j表示前已经到哪个位置了 
            {
                complex w(1, 0);//幂 
                for (int k = 0; k < mid; k++, w = w * Wn)//枚举左半部分 
                {
                    complex x = A[j + k], y = w * A[j + mid + k];//蝴蝶效应 
                    A[j + k] = x + y;
                    A[j + mid + k] = x - y;
                }
            }
        }
    }

}fft;
int ans[MAX_FFT];
int main()
{
    ios::sync_with_stdio(false);
    cin >> s1 >> s2; n = s1.size() - 1; m = s2.size() - 1;
    for (int i = n, j = 0; i >= 0; i--, j++)a[j].x = s1[i] - '0';
    for (int i = m, j = 0; i >= 0; i--, j++)b[j].x = s2[i] - '0';
   int len = n + m+2;
    fft.init(len);

    fft.fast_fast_tle(a, 1); //1表示从系数变为点值
    fft.fast_fast_tle(b, 1); //1表示从系数变为点值

    for (int i = 0; i <= fft.limit; i++)a[i] = a[i] * b[i];
    fft.fast_fast_tle(a, -1);  //-1表示从点值变为系数 



    for (int i = 0; i <= n + m; i++)
    {
        ans[i] += (a[i].x / fft.limit + 0.49);
        ans[i + 1] += ans[i] / 10;
        ans[i] %= 10;
    }
  
    while (!ans[len] && len > 0)len--;
    for (int i = len; i >= 0; i--)cout << ans[i]; cout << '\n';
    //for (int i = 0; i <= n + m; i++)printf("%d ", (int)(a[i].x / fft.limit + 0.5));
    return 0;
}