I hope I'm not duplicating!

I'm wondering how it is possible to find all roots of a polynomial of very high degree (100,1000,1000000, ...) numerically.

In all numerical methods, the polynomial is evaluated in a certain point at each iteration and is there any simple computer which can cope with calculations on a polynomial of degree 1000000 for instance?

Can somebody suggest some resources?


PS: I checked Computing roots of high degree polynomial numerically. and Finding all complex zeros of a high-degree polynomial but they don't have my answers.


Every computer can deal with $5^{1000000}$ if appropriately programmed to do so (and if it has enough memory to store all of the digits you need). Such numbers are too large for a computer to do arithmetic on in a single step, but then you just use multiple steps instead -- like you can multiply 5-digit numbers on paper by working digit by digit.

This takes longer time per operation than working with numbers that are small enough to be handled natively by the hardware -- but if you have a million-degree polynomial you need evaluated, you just have to wait the time it takes.

(Actually, given the numbers in the questions you link to, the millionth degree is probably out of reach presently for resource reasons).


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