P3338 [ZJOI2014]力

$$\begin{gathered} F_j=\sum _{i=1}^{j?1}\frac{q_i\times q_j}{(i?j{)}^2}?\sum _{i=j+1}^n\frac{q_i\times q_j}{(i?j{)}^2} \\ {E}_j=\sum _{i=1}^{j?1}\frac{q_i}{(i?j{)}^2}?\sum _{i=j+1}^n\frac{q_i}{(i?j{)}^2} \\ f(i)=q_i,g(i)=\frac{1}{i^2},f(0)=0,g(0)=0 \\ {E}_j=\sum _{i=0}^{j}f(i)g(j?i)?\sum _{i=j}^{n}f(i)g(i?j) \end{gathered}$$

前項是標準的多項式卷積，后項我們另考慮后項。

$\sum _{i=j}^{n}f(i)g(i?j)$，$T=i?j$，$i=T+j$，原式子$\sum _{i=0}^{n?j}f(i+j)g(i)$。

我們翻轉$f(i)$得到$f^{\prime }(n?i)=f(i)$，代入原式子$\sum _{i=0}^{n?j}{f}^{\prime }(n?(i+j))g(i)$。

$n?j=T$，$\sum _{i=0}^T{f}^{\prime }(T?i)g(i)$，也就是一個多項式卷積了。

#include <bits/stdc++.h>using namespace std;struct Complex {double r, i;Complex(double _r = 0, double _i = 0) : r(_r), i(_i) {} };Complex operator + (const Complex &a, const Complex &b) {return Complex(a.r + b.r, a.i + b.i); }Complex operator - (const Complex &a, const Complex &b) {return Complex(a.r - b.r, a.i - b.i); }Complex operator * (const Complex &a, const Complex &b) {return Complex(a.r * b.r - a.i * b.i, a.r * b.i + a.i * b.r); }Complex operator / (const Complex &a, const Complex &b) {return Complex((a.r * b.r + a.i * b.i) / (b.r * b.r + b.i * b.i), (a.i * b.r - a.r * b.i) / (b.r * b.r + b.i * b.i)); }typedef long long ll;const int N = 1e6 + 10;int r[N];Complex a[N], b[N], c[N];void get_r(int lim) {for (int i = 0; i < lim; i++) {r[i] = (i & 1) * (lim >> 1) + (r[i >> 1] >> 1);} }void FFT(Complex *f, int lim, int rev) {for (int i = 0; i < lim; i++) {if (i < r[i]) {swap(f[i], f[r[i]]);}}const double pi = acos(-1.0);for (int mid = 1; mid < lim; mid <<= 1) {Complex wn = Complex(cos(pi / mid), rev * sin(pi / mid));for (int len = mid << 1, cur = 0; cur < lim; cur += len) {Complex w = Complex(1, 0);for (int k = 0; k < mid; k++, w = w * wn) {Complex x = f[cur + k], y = w * f[cur + mid + k];f[cur + k] = x + y, f[cur + mid + k] = x - y;}}}if (rev == -1) {for (int i = 0; i < lim; i++) {f[i].r /= lim;}} }int main() {// freopen("in.txt", "r", stdin);// freopen("out.txt", "w", stdout);// ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);int n, lim = 1;scanf("%d", &n);for (int i = 1; i <= n; i++) {scanf("%lf", &a[i].r);b[n - i].r = a[i].r;c[i].r = 1.0 / i / i;}n <<= 1;while (lim <= n) {lim <<= 1;}get_r(lim);FFT(a, lim, 1);FFT(b, lim, 1);FFT(c, lim, 1);for (int i = 0; i < lim; i++) {a[i] = a[i] * c[i];b[i] = b[i] * c[i];}FFT(a, lim, -1);FFT(b, lim, -1);n >>= 1;for (int i = 1; i <= n; i++) {printf("%.3f\n", a[i].r - b[n - i].r);}return 0; }

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