

A Spreadsheet for finding Roots of Polynomials - karamazov
https://datanitro.com/blog/2013/7/9/spreadsheet_for_polynomial_roots/

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Someone
_" If you change a polynomial's coefficients slightly, its roots will only
change slight as well."_

If only:
[http://en.wikipedia.org/wiki/Wilkinson's_polynomial](http://en.wikipedia.org/wiki/Wilkinson's_polynomial)

~~~
karamazov
That's a great point, and an awesome article. In the post, I'm referring to a
continuous change versus a discontinuous one; with large coefficients or large
powers, you can cause a big shift with a very small change.

However, you can always find a chance small enough that it won't make a
substantial impact on the roots; this isn't the case with discontinuous
functions.

~~~
pfortuny
Ehm. The big (as in BIG BROTHER) problem is that mere continuity is a very
very weak property. The fact that "small" changes turn into other "small"
changes depends essentially on the "smallness".

For example (I know this is not a polynomial, but the function is related to
root-finding):

f(x) = x^(1/10)

which has x=0 as a root (f(0)=0).

Take a "small" increment of x, say x=0.01

f(0.01) = 0.63+

Which, to any reasonable human being is "quite large" with respect to the
increment of x.

This is even worse for Wilkinson's Polynomial, which is what the parent was
trying to convey, I guess.

I know I am referring just to a related problem but this is one of those days
for me...

That is: roots depend _continuously_ on the coefficients of a polynomial but
that _continuity_ is filled with very bad behaved constants all around dancing
and whirling to make the problem very very bad behaved.

EDIT: I think I have realised what I want to say.

Take P(x) = x^10, it has a (multiplicity 10 root at x=0). Then take Q(x) =
x^10 - .01 (which is a "small" deformation of P(x)). Then the roots of Q are
0.63+ and -0.63+, which are "huge" compared to 0, 0.01.

Unless I am still under this terrible drosiness...

~~~
stephencanon
The dependence of (real) roots on the coefficients of a polynomial isn’t even
continuous; an arbitrarily small change to the coefficients of a polynomial
with a double root will result in either the appearance of an extra root or in
the (real) root disappearing entirely.

In the complex plane the situation is much happier, but root-finding remains
extremely poorly conditioned for examples like Wilkonson’s.

~~~
pfortuny
Yes, yes, I know that, I was focusing on getting an example.

I think I should call it a day and reboot tomorrow...

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joshhertz
Section 3 "Roots are closely related to factoring a polynomial - for example,
the roots of x2 - 4 are 2 and 2, and you can write x2 - 4 as (x-2)(x-2). We
say that a root r has a multiplicity greater than one if (x-r) is a factor of
the polynomial more than once." This is incorrect. x2 - 4 = (x - 2)(x + 2).
Meaning the roots are 2 and -2. Also section 5, I'm not sure how you came up
with those answers. The roots are 2 and -1. I tried the spreadsheet, but can't
seem to get it to work.

~~~
karamazov
these are fixed. thanks for pointing them out!

what's wrong with the spreadsheet?

