How to handle rounding error?

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Gobi 2015-5-16

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回答： Walter Roberson 2015-5-16

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 =

ans =

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

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Answer 1

Walter Roberson 2015-5-16

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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