On computation of a power series root with arbitrary degree of convergence

Given a bivariate polynomial $f(x,y)$, let $\phi(y)$ be a power series root of $f(x,y)=0$ with respect to $x$, i.e., $\phi(y)$ is a function of $y$ such that $f(\phi(y),y)=0$. If $\phi(y)$ is analytic at $y=0$, then we have its power series expansion \begin{equation} \phi(y)=\alpha_{0}+\alpha_{1}y+\alpha_{2}y^{2}+\cdots+\alpha_{r}y^{r}+\cdots. \end{equation} Let $\phi^{(k)}(y)$ denote $\phi(y)$ truncated at $y^{k}$, i.e., \begin{equation} \phi^{(k)}(y)=\alpha_{0}+\alpha_{1}y+\alpha_{2}y^{2}+\cdots+\alpha_{k}y^{k}. \end{equation} Then, it is well known that, given initial value $\phi^{(0)}(y)=\alpha_{0}\in\mathbf{C}$, the symbolic Newton's method with the formula \begin{equation} \phi^{(2^{m}-1)}(y)\gets\phi^{(2^{m-1}-1)}(y) -\frac{f(\phi^{(2^{m-1}-1)}(y),y)}{\frac{\partial f}{\partial x}(\phi^{(2^{m-1}-1)}(y),y)} \quad (\mod y^{2^{m}}) \end{equation} computes $\phi^{(2^{m}-1)}(y)$ ($1\le m$) in (2) with quadratic convergence (the roots are computed in the order $\phi^{(0)}(y) \to \phi^{(2^{1}-1)}(y) \to \phi^{(2^{2}-1)}(y) \to \cdots \to \phi^{(2^{m}-1)}(y)$). References [1] and [3] indicate that the symbolic Newton's method can be generalized so that its convergence degree is an arbitrary integer $p$ where its roots are computed in the order $\phi^{(0)}(y) \to \phi^{(p-1)}(y) \to \phi^{(p^{2}-1)}(y) \to \cdots \to \phi^{(p^{m}-1)}(y)$. Although the high degree convergent formula in [1] and [3] requires fewer iterations than the symbolic Newton's method, it may not be efficient as expected, since one iteration of the formula requires more computations than one in the symbolic Newton's method. ¶ In this paper, we combine the polynomial evaluation method in [9] with the formula of arbitrary degree convergence and propose an algorithm that computes the above power series root $\phi^{(k)}(y)$. We analyze the complexity of the algorithm and give the number of multiplications/divisions required to compute a power series root in an explicit form. It is shown that when the degree of polynomial $f(x,y)$ is high, high degree convergent formula is advantageous over the symbolic Newton's method.