Re: All roots real for small degree polynomials

RobertMaas_at_YahooGroups.Com
Date: 08/03/04 

Date: Tue, 03 Aug 2004 00:28:58 -0700

> From: mckay@cs.concordia.ca (MCKAY john)
> What are the conditions for all the roots of a polynomial (degree
> small) to be real?

FIrst of all, are all the coefficients of the polynomial real?

Second, there's no such thing as roots of a polynomial, what you mean
to say are zeroes of the polynomial, or roots of the equation
polynomial=0.

Assuming all real coefficients, each such coefficient known to
unlimited precision and accuracy, are you seeking a numerical method
for answering the question whether the zeroes of the polynomial are all
real?

If the polynomial is square-free, there's an easy solution which is
recursive on the order of the polynomial: Isolate the real zeroes of
the derivative, then calculate the value of the original polynomial at
those points and check where they alternate in sign, thereby isolating
the real zeroes of the original polynomial. For degree n square-free
polynomial, there should be exactly n zeroes isolated if all zeroes are
real.

If the polynomial is not square-free, make it so by dividing by the GCD
of it and its derivative.

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