How to handle rounding error?

I want to check if the roots of a polynomial have absolute value <= 1. tried to handle roundoff error by using eps, but only eps does not work as follows:

p=[1 1 0 0 1 0 0 1 1];
abs(roots(p)) <= 1+eps
abs(roots(p)) <= 1+8*eps

The result:

ans =

     0
     0
     0
     0
     0
     0
     1
     1

ans =

     1
     1
     1
     1
     1
     1
     1
     1

So in usual, how should I write a code? Should I use something like 100*eps?

Answer 1

[Walter Roberson]

By the binomial theorem,

(a+b)^n = a^n + n*a^(n-1)*b + n*(n-1)/2 * a^(n-2)*b^2 + more terms

when b is very small compared to a, then b^2 will be even smaller, and when dealing with floating-point-roundoff-small indicates a term that can probably be neglected unless a is relatively large. So with round-off taken into consideration,

(a+b)^n <=> a^n + n*a^(n-1)*b

and when a is 1, then this naturally simplifies to 1 + n*b

You had an 8th order polynomial (your vector was length 9), so your n=8, and that is the source of your 8 for 8*eps