Newton's Method for Polynomial Powers

Newton’s Method for Polynomial Powers is an algorithm that computes $f(x{)}^z\mathrm{mod}{x}^n$ for a formal power series $f(x)$ such that $f(0)\ne 0$ in $\mathcal{O}(n\log n)$ time and $\mathcal{O}(n)$ space.

Algorithm

Lemma

$$f(x{)}^z=f(0{)}^z{e}^{z{\log }_e\frac{f(x)}{f(0)}}$$

Proof

$$\begin{array}{rl}f(x{)}^z& =f(0{)}^z{\left(\frac{f(x)}{f(0)}\right)}^z\\ & =f(0{)}^z{e}^{z{\log }_e\frac{f(x)}{f(0)}}\end{array}$$

0. Apply Newton’s Method for Polynomial Logarithms to find ${\log }_e\frac{f(x)}{f(0)}\mathrm{mod}{x}^n$.
1. Apply Newton’s Method for Polynomial Exponentiation to find $e^{z{\log }_e\frac{f(x)}{f(0)}}\mathrm{mod}{x}^n$.
2. Apply the lemma to find $f(x{)}^z\mathrm{mod}{x}^n$.

This algorithm solves the problem in $\mathcal{O}(n\log n)$ time and $\mathcal{O}(n)$ space.

std::vector<std::complex<double>> newton_pow(const std::vector<std::complex<double>> &a, std::complex<double> z, int n) {
	auto c = a[0];
 
	auto x = newton_log(std::ranges::to<std::vector>(std::views::iota(0, n) | std::views::transform([&](int i) -> std::complex<double> {
		return (i < int(a.size()) ? a[i] : 0) / c;
	})), n);
	for (int i = 0; i < n; i++) {
		x[i] *= z;
	}
	x = newton_exp(x, n);
	for (int i = 0; i < n; i++) {
		x[i] *= std::pow(c, z);
	}
	return x;
}