add lazy_exp and newton.hpp

Author: adamant-pwn

cp-algo/math/lazy_convolution.hpp

@@ -64,6 +64,22 @@ namespace cp_algo::math {
         cdq(1, n);
         return C;
     }
+
+    template<typename base>
+    auto lazy_exp(base p1, auto &&get_p, size_t n) {
+        // Q = exp P
+        // Q' = Q * P'
+        auto Qd = lazy_multiply<base>(base(1), p1, [&](auto &Q, auto &Pd, auto &Qd, size_t k) {
+            auto pdk = get_p(Qd, Pd, k + 1) * base(k + 1);
+            auto qk = Qd[k - 1] * small_inv<base>(k);
+            return std::pair{qk, pdk};
+        }, n);
+        for(size_t i = n - 1; i >= 1; i--) {
+            Qd[i] = Qd[i - 1] * small_inv<base>(i);
+        }
+        Qd[0] = base(1);
+        return Qd;
+    }
 }
 
 #endif // CP_ALGO_MATH_LAZY_MULTIPLY_HPP

cp-algo/math/newton.hpp

@@ -0,0 +1,22 @@
+#ifndef CP_ALGO_MATH_NEWTON_HPP
+#define CP_ALGO_MATH_NEWTON_HPP
+
+#include "poly.hpp"
+
+namespace cp_algo::math {
+    template<typename base>
+    auto newton(auto &&FFd, base f0, size_t n) {
+        using polyn = poly_t<base>;
+        polyn f = polyn(f0);
+        for(size_t len = 1; len < n; len *= 2) {
+            // f -= F(f) / F'(f)
+            auto [Ff, Fdf] = FFd(f, 2 * len);
+            Ff.div_xk_inplace(len);
+            Ff.mul_truncate(Fdf.inv_inplace(len), len);
+            f -= Ff.mul_xk_inplace(len);
+        }
+        return f.mod_xk_inplace(n);
+    }
+}
+
+#endif // CP_ALGO_MATH_NEWTON_HPP