From: Tri-solaris <trisolaris@126.com>
Date: Fri, 1 Mar 2019 15:06:02 +0000 (+0800)
Subject: feat: Update poly-inv.md
X-Git-Url: http://git.osdn.net/view?a=commitdiff_plain;h=995526472e1366ce4d6c15a15447d52590f438d0;p=oi-wiki%2Fmain.git

feat: Update poly-inv.md
---

diff --git a/docs/math/poly-inv.md b/docs/math/poly-inv.md
new file mode 100755
index 00000000..80ad1a1d
--- /dev/null
+++ b/docs/math/poly-inv.md
@@ -0,0 +1,75 @@
+# Description
+
+给定多项式 $f\left(x\right)$,求 $f^{-1}\left(x\right)$.
+
+# Methods
+
+## 倍增法
+
+首先,易知
+
+$$\left[x^{0}\right]f^{-1}\left(x\right)=\left(\left[x^{0}\right]f\left(x\right)\right)^{-1}$$
+
+假设现在已经求出了 $f\left(x\right)$ 在模 $x^{\left\lceil\frac{n}{2}\right\rceil}$ 意义下的逆元 $f^{-1}_{0}\left(x\right)$.
+有:
+
+$$\begin{aligned}
+	f\left(x\right)f^{-1}_{0}\left(x\right)&\equiv 1 &\pmod{x^{\left\lceil\frac{n}{2}\right\rceil}}\\
+	f\left(x\right)f^{-1}\left(x\right)&\equiv 1 &\pmod{x^{\left\lceil\frac{n}{2}\right\rceil}}\\
+	f^{-1}\left(x\right)-f^{-1}_{0}\left(x\right)&\equiv 0 &\pmod{x^{\left\lceil\frac{n}{2}\right\rceil}}
+\end{aligned}$$
+
+两边平方可得:
+
+$$f^{-2}\left(x\right)-2f^{-1}\left(x\right)f^{-1}_{0}\left(x\right)+f^{-2}_{0}\left(x\right)\equiv 0 \pmod{x^{n}}$$
+
+两边同乘 $f\left(x\right)$ 并移项可得:
+
+$$f^{-1}\left(x\right)\equiv f^{-1}_{0}\left(x\right)\left(2-f\left(x\right)f^{-1}_{0}\left(x\right)\right) \pmod{x^{n}}$$
+
+递归计算即可.
+
+时间复杂度
+
+$$T\left(n\right)=T\left(\frac{n}{2}\right)+O\left(n\log{n}\right)=O\left(n\log{n}\right)$$
+
+## Newton's Method
+
+参见 [**Newton's Method**](../poly-newton/#inv).
+
+# Code
+
+??? " `poly-inv.cpp` "
+
+    ```cpp
+    inline void cp(const Z*const&sl,const Z*const&sr,Z*const&dl,Z*const&dr){
+        std::copy(sl,sr,dl);
+        if(sr-sl<dr-dl)
+            std::fill(dl+(sr-sl),dr,0);
+    }
+
+    void polyinv(const poly&h,poly&f,const int&n){
+        static poly_t inv_t;
+        std::fill(f,f+n+n,0);
+        f[0]=fpow(h[0],p-2);
+        for(int t=1;(1<<t)<=n;++t){
+            const int deg1=1<<t,deg2=deg1<<1;
+            cp(h,h+deg1,inv_t,inv_t+deg2);
+
+            DFT(f,deg2);DFT(inv_t,deg2);
+            for(int i=0;i!=deg2;++i)
+                f[i]=(mZ)f[i]*sub(2ll,(mZ)inv_t[i]*f[i]%p)%p;
+            IDFT(f,deg2);
+
+            std::fill(f+deg1,f+deg2,0);
+        }
+    }
+    ```
+
+
+
+# Examples
+
+1. 有标号简单无向连通图计数:「BZOJ 3456」城市规划
+
+
diff --git a/mkdocs.yml b/mkdocs.yml
index b5abe238..c07ea49b 100644
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -122,6 +122,7 @@ nav:
       - 快速傅里叶变换: math/fft.md
       - 快速数论变换: math/ntt.md
       - 快速沃尔什变换: math/fwt.md
+      - 多项式求逆: math/poly-inv.md
     - 组合数学:
       - 排列组合: math/combination.md
       - 卡特兰数: math/catalan.md