Weird stuff near zero

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cameron432
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Weird stuff near zero

Postby cameron432 » Mon Jan 24, 2011 1:55 pm UTC

Hey guys, I've got a question. I've always been fascinated by this. Why, in a normal polynomial, does all of the weird stuff (the curving and changing directions) happen relatively near x=0?

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Yakk
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Re: Weird stuff near zero

Postby Yakk » Mon Jan 24, 2011 2:07 pm UTC

Have you done any calculus?
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Re: Weird stuff near zero

Postby cameron432 » Mon Jan 24, 2011 2:14 pm UTC

Yeah, I'm in Calc 3. I feel like because you said that, this answer is going to be REALLY obvious....
...
And never mind. Thanks. Makes sense now.

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Re: Weird stuff near zero

Postby Yakk » Mon Jan 24, 2011 2:25 pm UTC

There is that. On top of that, O-notation gives a hint -- the behavior of a function as x->infinity is dominated by the highest coefficient.

This domination is basically complete for a function f(x) := product( a_i x^i ) of degree n when |x| >= sum(a_i over i<n)/a_n -- in fact earlier, but I'm talking about sufficient -- because by that point, the highest term in the polynomial has grown to at least the size of the rest of them combined!

And it proceeds to grow faster than the rest.

With relatively small coefficients and ratios between coefficients, this means that the point where the top coefficient completely dominates is relatively close to zero.
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Re: Weird stuff near zero

Postby undecim » Mon Jan 24, 2011 4:31 pm UTC

I've always thought of zero as very similar to +/- infinity. For example, look at 1/x as x->0. Looking at the positive side of the graph, the naïve (e.g. newbie) mathematician might say that 0 is the reciprocal of infinity

You can use that to visualize a function in [x,infinity]. For example, to see the hyperbolic tangent function go to infinity, try graphing tanh(1/x). Instead of starting at (0,0) and limiting to (infinity,1), here it limits at (0,1), and (infinity, 0).

Do this with sine, and you get an infinite number of oscillations near 0.
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emwilso
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Re: Weird stuff near zero

Postby emwilso » Tue Jan 25, 2011 2:14 pm UTC

Probably because when dealing with polynomials we generally use nice numbers for our coefficients. Imagine if your professor gave you a polynomial that had roots above 1000 it wouldn't test your knowledge of the material but force you to do redundant calculations.

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Re: Weird stuff near zero

Postby rauni » Tue Jan 25, 2011 2:25 pm UTC

What do You mean, "normal polynom"? x,x^2,x^3, ...?

If You look at odd powers, x,x^3, x^5,... then it is obvious that if x=0, then x^odd = 0. But negative number to odd power is negative, and positive to positive, so one part will go up and other will go down. And because, for example, (-x)^3=-(x^3), the left side will start to go to -infinity with the same tempo as right side goes to +infinity.

About even powers it is similar, only that both sides go towards the same infinity.

This makes sense to me to think about it like this. Please let me know if I was clear on my answer.

On the other hand, if You take some other polynom, it may not have anything special happening in point 0. For example, (x-2)^2=x^2-4*x+4. It is the same x^2, only the whole parabole has been moved 2 to the right. (can You understand why?). There the "magic" of curving and changing direction happens at point 2.

Rauni

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Re: Weird stuff near zero

Postby gorcee » Tue Jan 25, 2011 3:06 pm UTC

rauni wrote:What do You mean, "normal polynom"? x,x^2,x^3, ...?

If You look at odd powers, x,x^3, x^5,... then it is obvious that if x=0, then x^odd = 0. But negative number to odd power is negative, and positive to positive, so one part will go up and other will go down. And because, for example, (-x)^3=-(x^3), the left side will start to go to -infinity with the same tempo as right side goes to +infinity.

About even powers it is similar, only that both sides go towards the same infinity.

This makes sense to me to think about it like this. Please let me know if I was clear on my answer.

On the other hand, if You take some other polynom, it may not have anything special happening in point 0. For example, (x-2)^2=x^2-4*x+4. It is the same x^2, only the whole parabole has been moved 2 to the right. (can You understand why?). There the "magic" of curving and changing direction happens at point 2.

Rauni


You can shift any polynomial P(x) by taking P(y) where y = x - k. However, go ahead and expand all the (y - k) ^ m terms, group like powers, then you'll still end up with P(y) = c1 y^m + c2 y^m-1 + ...

The point is, for any polynomial, all of the "interesting stuff" happens within a ball around 0 (this argument still holds if you shift things). In order to get things sufficiently away from zero, you have to choose coefficients or shifting factors sufficiently large that the leading term in the polynomial does not dominate. It is uncommon to see such coefficients in practice.

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Re: Weird stuff near zero

Postby rauni » Tue Jan 25, 2011 3:18 pm UTC

gorcee wrote:
rauni wrote:What do You mean, "normal polynom"? x,x^2,x^3, ...?
...........
Rauni

You can shift any polynomial P(x) by taking P(y) where y = x - k. However, go ahead and expand all the (y - k) ^ m terms, group like powers, then you'll still end up with P(y) = c1 y^m + c2 y^m-1 + ...

The point is, for any polynomial, all of the "interesting stuff" happens within a ball around 0 (this argument still holds if you shift things). In order to get things sufficiently away from zero, you have to choose coefficients or shifting factors sufficiently large that the leading term in the polynomial does not dominate. It is uncommon to see such coefficients in practice.


It seems I understood the question wrong. I did not read the word "relatively", I now understand that this may also mean, for example, 2.

cameron432 wrote:Hey guys, I've got a question. I've always been fascinated by this. Why, in a normal polynomial, does all of the weird stuff (the curving and changing directions) happen relatively near x=0?


Then I agree with what emwilso said, that
emwilso wrote:Probably because when dealing with polynomials we generally use nice numbers for our coefficients. Imagine if your professor gave you a polynomial that had roots above 1000 it wouldn't test your knowledge of the material but force you to do redundant calculations.


It would be interesting to do some calculations however: if we take nth order polynom with coefficients absolutely smaller than some fixated constant, what is the average root distance from point 0?

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Re: Weird stuff near zero

Postby gorcee » Tue Jan 25, 2011 6:04 pm UTC

rauni wrote:It would be interesting to do some calculations however: if we take nth order polynom with coefficients absolutely smaller than some fixated constant, what is the average root distance from point 0?


This is probably not too hard to do. I seem to remember coming across some work on computing the maximum root of a polynomial. You could then insert some distribution on your relevant coefficients and compute the expected value of the maximal root, reduce the order of the polynomial, and repeat.

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Re: Weird stuff near zero

Postby Yakk » Tue Jan 25, 2011 6:09 pm UTC

rauni wrote:It would be interesting to do some calculations however: if we take nth order polynom with coefficients absolutely smaller than some fixated constant, what is the average root distance from point 0?

Lets look at 1st order polynomials: a x + b

The root is then at x = -b/a

If a and b are uniform random variables over the interval [-epsilon, +epsilon], this gives us a new non-uniform random variable for where the root is.

We then take the absolute value of this new random variable, and calculate the expectation. I don't want to sound hyperbolic, but this might not actually model what you want to model. :)
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Re: Weird stuff near zero

Postby rauni » Tue Jan 25, 2011 6:28 pm UTC

Yakk wrote:
rauni wrote:It would be interesting to do some calculations however: if we take nth order polynom with coefficients absolutely smaller than some fixated constant, what is the average root distance from point 0?

Lets look at 1st order polynomials: a x + b

The root is then at x = -b/a

If a and b are uniform random variables over the interval [-epsilon, +epsilon], this gives us a new non-uniform random variable for where the root is.

We then take the absolute value of this new random variable, and calculate the expectation. I don't want to sound hyperbolic, but this might not actually model what you want to model. :)


No, that is exactly what I meant :) then find the same thing for n=2, n=3, ... , and then we can see if this value converges when n goes to infinity.

Although, I am not exactly sure how to go farther from case n=1... :)
gorcee wrote:This is probably not too hard to do. I seem to remember coming across some work on computing the maximum root of a polynomial. You could then insert some distribution on your relevant coefficients and compute the expected value of the maximal root, reduce the order of the polynomial, and repeat.


Can You provide a link?

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Re: Weird stuff near zero

Postby Yakk » Tue Jan 25, 2011 6:36 pm UTC

Ah. Well, to be more clear, the answer is "it diverges".

Take the region with |b| > epsilon/2. Thus the integral of | b/a | is at least the integral of | epsilon / a | over the region [0,epsilon] (the /2 and the plus-minus copies cancel out). And integrating 1/x around 0 doesn't work well if you want a finite answer.

Unless I made a mistake, which is reasonably likely.
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Re: Weird stuff near zero

Postby gorcee » Tue Jan 25, 2011 7:33 pm UTC

rauni wrote:Can You provide a link?


It's severely non-trivial.

This article talks about computing all the roots using eigenvalue methods. http://portal.acm.org/citation.cfm?id=3 ... N=42432955

Most eigenvalue methods are based on deflation, ie, computing the largest eigenvalue first, and systematically reducing the rank of the problem. The problem of computing the average distance of all the roots would be equivalent to computing the average magnitude of the eigenvalues of a matrix. You can probably substitute in random variables for the polynomial coefficients in the formulation of the problem, and then operate on spectral expansions (ie, Weiner-Askey polynomial chaos) of the random variables to eventually compute a result.

This result generalizes to complex roots, but it does mention that there are many, many articles discussing this problem: http://books.google.com/books?id=4PMqxw ... &q&f=false

I think it is an interesting problem, but I doubt it would be simple to solve.

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Re: Weird stuff near zero

Postby gmalivuk » Tue Jan 25, 2011 10:59 pm UTC

Yakk wrote:
rauni wrote:It would be interesting to do some calculations however: if we take nth order polynom with coefficients absolutely smaller than some fixated constant, what is the average root distance from point 0?

Lets look at 1st order polynomials: a x + b

The root is then at x = -b/a

If a and b are uniform random variables over the interval [-epsilon, +epsilon], this gives us a new non-uniform random variable for where the root is.

We then take the absolute value of this new random variable, and calculate the expectation.
Let Y=abs(x), then Y=abs(b)/abs(a), and the absolute values of b and a are uniformly distributed between 0 and epsilon.

P(Y>y) = P(abs(b)/abs(a)>y)=P(abs(b)>y*abs(a)), which is the normalized area of the square [0,epsilon]2 that's above the line through the origin with slope y. That is, This probability is the area above that line and within [0,1]2. For y<1 this is 1-y/2, and for y>1 it's 1/(2y). Since the expected value can be found by integrating P(Y>y) from y=0 to infinity, this indeed diverges, and the expected value is infinite.
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However, I suspect that this is no longer the case with higher degree polynomials, but I don't particularly care to figure it out explicitly.
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Re: Weird stuff near zero

Postby gorcee » Wed Jan 26, 2011 2:19 pm UTC

I think it's a little disingenuous to include 0 in our analysis for the distributions of a and b. Allowing a to be zero generates an infinite number of polynomials with no roots (y = b), and allowing b to be zero generates an infinite number of polynomials whose root is exactly 0.

I think the problem could be fixed by explicitly requiring nonzero coefficients, or possibly restating the problem to one of monic polynomials.

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Re: Weird stuff near zero

Postby Qaanol » Wed Jan 26, 2011 2:36 pm UTC

cameron432 wrote:Hey guys, I've got a question. I've always been fascinated by this. Why, in a normal polynomial, does all of the weird stuff (the curving and changing directions) happen relatively near x=0?

What made you pick out x=0 as special? If you hand me any polynomial in x, I’ll plug in y = x+1, expand it out, and hand you right back a polynomial in y. Anything you said happens near zero now happens near -1.
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Re: Weird stuff near zero

Postby gmalivuk » Wed Jan 26, 2011 2:47 pm UTC

gorcee wrote:I think it's a little disingenuous to include 0 in our analysis for the distributions of a and b. Allowing a to be zero generates an infinite number of polynomials with no roots (y = b), and allowing b to be zero generates an infinite number of polynomials whose root is exactly 0.
Both sets have measure zero in our probability distribution, so they don't really affect anything.

So fine, let's require nonzero coefficients.
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Re: Weird stuff near zero

Postby imatrendytotebag » Thu Jan 27, 2011 12:12 pm UTC

Maybe we can get a meaningful answer by requiring the polynomial to be monic, ie the leading coefficient is 1. Because any nth degree polynomial that is not monic has the same roots if you divide all coefficients by the leading coefficient, so by making the leading coefficient very small you can get rather large roots.

For 1st degree polynomials with leading coefficient 1 and constant term evenly distributed on [-epsilon,epsilon], the average distance of the root from 0 is epsilon/2.

I don't know about 2nd degree polynomials, but I imagine it's a bit more complicated. It is worth noting, however, that if the x coefficient and constant term are bounded by epsilon, the absolute value of the roots are less than epsilon + 1. In other words, the average distance should exist.

In second degree polynomials (and higher), we have two options. We can either restrict to polynomials with real roots, and let b and c be uniformly distributed given that the roots are real, or we let b and c be anything and use the complex distance formula. If we're in the latter scenario, it might be interesting to instead let b and c be complex numbers uniformly distributed on the circle of radius epsilon about 0.
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Re: Weird stuff near zero

Postby rauni » Thu Jan 27, 2011 12:55 pm UTC

I found an way to prove that for any n-th order polynom with bounded coefficents (all coeffiecents < epsilon), the expected value of absolute average of root is infinite.

First, let polynom be in form a(x-x1)(x-x2)..(x-xn)=ax^n-(x1+x2+...+xn)x^(n-1)+...

Here we see that sum of roots is always -b/a. Let sum of roots be z.

This is not exactly what we need, because we need sum of absolutes of roots, but we have sum of roots. However, we can use to prove this that average of absolutes of roots is infinite.

Always |z|=|x1+x2+...+xn|<=|x1|+|x2|+...+|xn|. So, |z/n|<=(|x1|+...+|xn|)/n.

As shown by Yakk and gmalivuk, expected value for |z| is infinite, therefore so is |z/n| and expected value of average of absolutes of roots.

QED?

If this is correct, then we can do as imatrendytotebag said: try monic polynoms.

imatrendytotebag wrote:Maybe we can get a meaningful answer by requiring the polynomial to be monic, ie the leading coefficient is 1. Because any nth degree polynomial that is not monic has the same roots if you divide all coefficients by the leading coefficient, so by making the leading coefficient very small you can get rather large roots.

For 1st degree polynomials with leading coefficient 1 and constant term evenly distributed on [-epsilon,epsilon], the average distance of the root from 0 is epsilon/2.

I don't know about 2nd degree polynomials, but I imagine it's a bit more complicated. It is worth noting, however, that if the x coefficient and constant term are bounded by epsilon, the absolute value of the roots are less than epsilon + 1. In other words, the average distance should exist.

In second degree polynomials (and higher), we have two options. We can either restrict to polynomials with real roots, and let b and c be uniformly distributed given that the roots are real, or we let b and c be anything and use the complex distance formula. If we're in the latter scenario, it might be interesting to instead let b and c be complex numbers uniformly distributed on the circle of radius epsilon about 0.


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