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More on Bounds on the Roots of Polynomials

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In an earlier article/post (reference 1) I showed the Cauchy upper bound for the absolute cvalue of the roots of a monic polynomial. The restriction to a monic polynomial was purly for convienience, but here we will require an arbitary polynomial  of degree n so the Cauchy lower bound is: If  x is a root of the polynomial:

P(x)=a_nx^n+a_{n-1}x^{n-1}+ ... + a_0

then:

|x| \le 1+ \frac{\max_{i=0, .. , n-1} |a_i|}{|a_n| }

Now I will extend this to give a lower bound for the absolute value of the non-zero roots of P(x) Without any real loss of generality we can assume that both a_n and a_0 \ne 0 (if the lowest power of  x appearing in p(x) is \rho \ne 0  we divide through by x^{\rho} to get a polynomial of the required form and whos non-zero roots are the same as the polynomial we started with).

Let B(a_0, a_1, ..., a_n) be an an upper bound on the ansolute value of roota of the ploynomial: P(x)=a_nx^n+a_{n-1}x^{n-1}+ ... + a_0 where a_n and a_0 are both non-zero.  Now let x be a root of P(x), so:

P(x)=a_nx^n+a_{n-1}x^{n-1}+ ... + a_0=0

Now divide through by x^n and put y=1/x, so we have:

Q(y)=a_0 y^n + a_1 y^{n-1}+ ... + a_n=0

Hence:

|y| \le B(a_n, a_1, ..., a_0)

or:

|x| \ge 1/B(a_n, a_1, ..., a_0) .

So combinining this lower bound with the upper bound we have for any n-th degree polynomial with only non-zero roots for any root x:

\frac{1}{B(a_n, a_{n-1}, ..., a_0) } \le |x| \le B(a_0, a_1, ..., a_n)

and for the Cauchy bound we have:

B(a_0, a_1, ... , a_n) =1+\frac{\max_{i=0, .. , n-1} |a_i|}{|a_n| }

So:

\frac{|a_0|}{|a_0|+\max_{i=1,..,n}|a_i|}\le |x| \le \frac{|a_n|+\max_{i=0,..,n-1}|a_i|}{|a_n|}

References:

1. https://captainblack.wordpress.com/2009/03/08/cauchys-upper-bound-for-the-roots-of-a-polynomial/

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Written by CaptainBlack

April 27, 2009 at 07:10

Posted in Maths and Stuff

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