Inverting a monotone polynomial

I have a polynomial of degree 5 $$f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5 x^5$$ that is strictly increasing (the derivative is always greater than zero). I would like to (approximately) invert this polynomial, i.e. find $$f^{-1}$$ such that $$f^{-1}(f(x)) = x$$. I know that $$f^{-1}$$ is unique and well-defined on all of $$\mathbb{R}$$ because of monotonicty of $$f(x)$$.

One possible solution that I'm aware of is to compute the inverse $$f^{-1}(y)$$ by finding the real root of the polynomial $$p(x) = f(x) - y$$ using e.g. Newton's method. $$f(x) = y \iff f(x) - y = 0$$ Do there exists alternative non-iterative approaches for (approximately) solving this problem? Is it possible to approximate $$f^{-1}$$ given the coefficients $$a_i$$?

• An issue is that the inverse function need not be a polynomial - for example if you take $f(x)=x^2$, then its inverse is $f^{-1}(x)=\sqrt{x}$. – asdf Jul 8 at 9:00
• math.stackexchange.com/questions/701061/… – Nurator Jul 8 at 9:26
• @Nurator, thank you, I am aware of that question. However, the answers there mostly talk about (1) existence of the inverse (which definitely exists in my case) and (2) iterative methods for finding the inverse – Alexander Shchur Jul 8 at 10:32
• Can you specify perhaps what kind of approximation you want, e.g. by a polynomial, in what regime e.g. x large, x small? – Calvin Khor Jul 8 at 10:37
• Approximation in terms of any closed-form expression (as defined in en.wikipedia.org/wiki/Closed-form_expression) would be great. I am concerned with "small" values of $x$ (in the range (-10, 10)) – Alexander Shchur Jul 8 at 10:53

Given a function : $$y(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5 x^5$$ such as $$x$$ and $$y$$ be real and the relationship be one-to-one, the inverse function $$x(y)$$ is the real root of the equation : $$A_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5 x^5=0\quad\text{where}\quad A_0=(a_0-y)$$ This is the quintic equation : http://mathworld.wolfram.com/QuinticEquation.html

In general (except for some particular values of the coefficients) the analytical solving for $$x$$ is not possible in terms of a finite number of elementary functions.

So, don't expect a non-iterative approche if you exclue the use of a convenient special function.

In the present case, the special functions involved are the Jacobi theta functions : http://mathworld.wolfram.com/JacobiThetaFunctions.html

This is an arduous analytical calculus. See the formal solution in http://mathworld.wolfram.com/QuinticEquation.html

On a practical viewpoint the use of numerical calculus is much simpler, but of course it doesn't satisfy your wish of non-iterative method.

• My hope was that even though quintic equations don't have closed-form roots in general, it would still be possible to find them in a relatively simple way due to monotonicity, (and hence uniqueness of the real root) in my special case. However, if that's not possible, then my question is answered I guess. – Alexander Shchur Jul 8 at 13:25