How approximate a sextic polynomial to a lower degree polynomial

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Discussion Overview

The discussion revolves around finding an approximate solution for a sextic polynomial derived from a nonlinear PDE related to waves. Participants explore methods to reduce the polynomial's degree to make it more manageable for symbolic root-finding.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Romik presents a sextic polynomial and seeks to approximate it with a lower degree polynomial for easier symbolic root finding.
  • One participant suggests dividing by \( u^6 \) to reformulate the problem and proposes using a Taylor expansion for an analytic approximation.
  • Romik clarifies that the goal is to reduce the polynomial from 6th to 4th degree for exact solutions.
  • Another participant discusses the relationship between the biggest root of the original equation and the smallest root of a modified equation, suggesting a Taylor approximation around the origin.
  • A participant proposes substitutions to simplify the polynomial, leading to a potentially simpler equation that could be solved for the variables involved.
  • Romik inquires about applying Newton's method for symbolic roots after truncating the polynomial to 4th degree.
  • A later reply indicates that with the polynomial truncated to 4th degree, analytic solutions are available, questioning the necessity of Newton's method.
  • Another participant explains how to apply Newton's method symbolically, using the function and its derivative to approximate the root.

Areas of Agreement / Disagreement

Participants express various methods for approximating the polynomial, but there is no consensus on the best approach or the necessity of using Newton's method after truncation.

Contextual Notes

Limitations include the dependence on the specific parameters of the polynomial and the unresolved nature of the substitutions proposed. The discussion also highlights the symbolic nature of the roots, which complicates the application of numerical methods.

Romik
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Hi all,

I have been stopped by a sextic (6th degree) polynomial in my research. I need to find the biggest positive root for this polynomial symbolically, and since its impassible in general, I came up with this idea, maybe there is a way to approximate this polynomial by a lower degree polynomial which is solvable.


κ2/112 (A2 ) u6+κ2/16 (A2 ) u5+κ2/20 (1/2 B2+3 A2 ) u4+κ2/8 (A2+B2 ) u3-((ω2-B2 κ2)/6) u2+ν2 κ2 ω2=0

this polynomial is come from a nonlinear PDE related to waves.
κ, A, B, v, ω , u are not constant.

I appreciate any helpful comment or solution.
Thanks,
Romik
 
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Divided by u6, this is a polynomial of 6th order in (1/u), where you look for the smallest positive root. Depending on the parameters, a taylor expansion or something similar might give some reasonable analytic approximation.
 
Thanks for the reply,
biggest or smallest positive root, that's not the main issue here, I need to find an approximate root based on variables, reduce from 6th degree to let's say 4th degree which I can solve it exactly.
 
Well, the biggest root in your original equation would be the smallest root in my modified equation.
On second thought, my idea with a taylor approximation around the origin would simply neglect the absolute term. The remaining polynomial can be expressed as u^2 P(u) where P has order 4, so there are analytic solutions. It might be interesting to improve this approximation with one or two steps of Newton afterwards ;).
 
I don't know if this would help, but... you could try the substitutions\begin{align*}<br /> x &amp;= A^2 u^2 \\<br /> y &amp;= B^2 u^2 \\<br /> z &amp;= \omega^2 v^2 \\<br /> s &amp;= \omega^2 u^2<br /> \end{align*}to obtain the possibly simpler equation<br /> \frac {\kappa^2 x} {112} u^4 + \frac {\kappa^2 x} {16} u^3 + \left( \frac {\kappa^2 (6x+y)} {40} \right)u^2<br /> + \frac{\kappa^2 (x+y)} 8 u - \frac{s-\kappa^2 y} 6 + \kappa^2 z = 0<br />
If you somehow manage to obtain values for u,x,y,z,s, then \omega = \pm\sqrt{\displaystyle\frac s {u^2}}, and the values for A,B,v can be solved for similarly.

(I was trying to put also \kappa into the substitutions for x,y,z, but then I can't find the original variables back. Unless you have an extra constraint on them.)
 
thanks mfb for you helpful comments.
can you explain more about Newton method, how could I apply it on my equation?
I use Mathematica! with Series function, I am able to truncate my original polynomial to 4th degree, now how should I apply Newton since I don't have numerical root and my roots are symbolical?

thank you Dodo for your reply, did you know you put your 666th post on this thread? So good luck to me :D
 
I am able to truncate my original polynomial to 4th degree
Great, that has analytic solutions, and you don't need Newton.

now how should I apply Newton since I don't have numerical root and my roots are symbolical?
Let x be the approximate position of the root, f(x) be the function value there and f'(x) its derivative. Both f(x) and f'(x) are easy to express symbolically. A (hopefully) better approximation for the root is then given by x-f(x)/f'(x).