Computation of the ultraradical

Question

The ultraradical of a real number, also called the Bring radical, is the unique real solution of the quintic equation

$$y=x^5+x.$$

This function is used in the resolution of general quintic equations, by reducing them to this particular form.

But are there specific methods to efficiently compute it numerically for arbitrary values of $x$ ?

Obviously, for small $y$, $x\approx y$, and for large $y$, $x\approx\sqrt[5]y$. But what about intermediate values ? Anything more specific than Newton ?

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Update:

After numerical tests, I observe that Newton's method, taken with the initial approximation $x=y$ or $x=\sqrt[5]y$ takes a maximum of $5$ iterations to reach $10^{-7}$ relative error o $x$. The worst cases are between $0.7$ and $3.5$.

This is already pretty good. I am now targeting a reduction to $3$ iterations by using a better initial approximation.

Here is a plot of a correction factor that should be introduced to improve the initial estimate:

Answer 1

Let the solution for $x$ be denoted by $\newcommand{BR}{\operatorname{BR}}\BR(y)$. We have by Lagrange inversion theorem (see link for more general case), for $|y|<4\cdot5^{-9/5}$,

$$\BR(y)=\sum _{{k=0}}^{\infty }{\binom {5k}{k}}{\frac {(-1)^{k}y^{{4k+1}}}{4k+1}}=y-y^{5}+5y^{9}-35y^{{13}}+\dots$$

On the other hand, a fixed point approach won't hurt. For large $|x|>1$, we should use the iteration

$$x_{n+1}=\sqrt[5]{y-x_n}$$

And for small $|x|<1$, we should use the iteration

$$x_{n+1}=y-x_n^5$$

Answer 2

To cover the transition interval: Choose a rational $\xi$ (e.g., $\xi=1$), and put $\eta:=\xi+\xi^5$. You then know that $b(\eta)=\xi$ and can easily determine the first few coefficients of the Taylor expansion of $b$ at $\eta$. Use this expansion to find a starting value for the Newton iteration.

An example: Letting $\xi=1$ you have $\eta=2$ and then $$b(y)\approx1+{1\over6}(y-2)-{5\over108}(y-2)^2\qquad(y\approx 2)\ .$$