Do there exist closed form approximating expressions for the roots of an nth order polynomial?
You won't find an expression over radicals for roots of polynomials above order 4, but you are asking about approximating roots without using iterations.
There are many approaches - depending on how good you need the approximation to be.
You have to do some of this in order to get the 1st approximation for the iterative approaches to work quickly.
AFAIK: there is no general approach for all polynomials - with computers, iterative approaches are fast and convenient so you don't have to be very accurate.
It is because I have an expression for a physical quantities, which depends on the roots of an nth order polynomial. I want to see if I can get an approximate closed form expression and to do that I need an approximate closed form for the root.
You may have heard of this chap called Galois. He proved that a general formula using arithmetic and radicals for solving for the roots of polynomials of degree greater than or equal to 5 was impractical, if not impossible.
It's like trying to go faster than the speed of light: all sorts of wonderful things could happen if this were possible, but alas, it is impossible. It is the same situation with finding the roots to your polynomial: numerically or not at all.
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