How approximate a sextic polynomial to a lower degree polynomial

AI Thread Summary
The discussion focuses on approximating a sextic polynomial to find its largest positive root symbolically. The original polynomial arises from a nonlinear PDE related to waves, and the goal is to reduce it to a solvable fourth-degree polynomial. A Taylor expansion is suggested as a method for approximation, while also considering substitutions to simplify the equation. The Newton method is mentioned as a potential technique for refining the root approximation, with symbolic expressions for the function and its derivative. Overall, the conversation emphasizes finding analytic solutions and improving approximations for the polynomial's roots.
Romik
Messages
13
Reaction score
0
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 ) u62/16 (A2 ) u52/20 (1/2 B2+3 A2 ) u42/8 (A2+B2 ) u3-((ω2-B2 κ2)/6) u22 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
 
Mathematics news on Phys.org
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).
 
Back
Top