Approximating unsolvable recursion relations

  • Context: Graduate 
  • Thread starter Thread starter Hoplite
  • Start date Start date
  • Tags Tags
    Recursion Relations
Click For Summary

Discussion Overview

The discussion revolves around a complicated recursion relation that is believed to be unsolvable in closed form. Participants explore methods for approximating the sum \( S(z) = \sum_{k=0}^\infty \xi_k z^k \) derived from the recursion, while addressing constraints on the series and potential solutions.

Discussion Character

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

Main Points Raised

  • One participant asserts that the recursion relation is unsolvable in closed form and presents specific constraints on the series.
  • Another participant suggests that the recursion may be solvable by taking differences of consecutive equations and using a geometric series as a trial solution, though they express uncertainty about the polynomial's degree.
  • A participant questions the interpretation of the sums of the series, initially suggesting that \( S \) should equal 2, but later corrects themselves.
  • One participant introduces the concept of the z-transform, proposing a related differential equation and seeking solutions, while noting that existing resources may only apply under special constraints.
  • Another participant comments on the rapid convergence of the series, suggesting that dealing directly with the recursive relation may be sufficient and questioning the necessity of a closed form.
  • A participant confirms the form of the differential equation related to the z-transform and mentions inhomogeneous boundary conditions.
  • One participant apologizes for any confusion caused and references a special case from a previous source that may be helpful.

Areas of Agreement / Disagreement

Participants express differing views on the solvability of the recursion relation and the necessity of a closed form solution. There is no consensus on the best approach to approximate the sum or the implications of the constraints presented.

Contextual Notes

Participants highlight various assumptions and conditions related to the recursion and the series, including the nature of the constraints and the applicability of certain methods. The discussion remains open-ended with unresolved mathematical steps.

Hoplite
Messages
50
Reaction score
0
I have a complicated recursion replation, which I'm sure is unsolvable. (By "unsolvable" I mean that there is no closed form solution expressing \xi_1, \xi_2, \xi_3, etc. in terms of \xi_0.) It goes

\frac{(k+4)!}{k!}\xi_{k+4} +K_1 (k+2)(k+1)\xi_{k+2}+ [ K_2 k(k-1) +K_3] \xi_{k} +K_4 \xi_{k-2} =0,

for k=0,1,2,3..., where K_1, K_2, K_3 and K_4 are constants. However, what I do know about the \xi_k's is that

\sum_{k_{even}} \xi_{k}=\sum_{k_{odd}} \xi_k=1,

and that

\sum_{k_{even}}k \xi_{k}=\sum_{k_{odd}}k \xi_k=0.

\xi_k is also zero for all k<0. What I want to do is produce an approximation of the sum for

S(z) =\sum_{k=0}^\infty \xi_k z^k.

Does anyone have any idea how to do this?
 
Last edited:
Physics news on Phys.org
If I remember correctly that type of recursion relation is solvable. Somehow you take differences of consecutive equations until you have
\sum_1^N a_i\xi_{k+i}=0
and then you use a geometric series as the trial solution. Not sure if the degree of that polynomial becomes too large.

But maybe the contraints help.
 
Hoplite said:
\sum_{k_{even}} \xi_{k}=\sum_{k_{odd}} \xi_k=1,
What do you mean? Isn't S=2 then?
 
Gerenuk said:
What do you mean? Isn't S=2 then?
Oops, sorry. I had the wrong equation for S. I've fixed it now.
 
Using something which I think is called z-transform you get
S''''+(a+bx^2)S''+(c+dx^2)S=e_5x^5+\dotsb+e_0
I guess that's your initial problem in reverse :)
Anyone knows how to solve this? I've found
http://eqworld.ipmnet.ru/en/solutions/ode/ode0406.pdf
but that only works for a special contraints on the parameters.
 
Last edited:
This series will converge really quickly - this is definitely an entire function. It's probably just as good to deal with the recursive relation. Even if you had a closed form, it wouldn't help you to evaluate the function.

Do you need a closed form, or do you just think you need it? For the purpose of it being a solution to an ODE, this is just as good.
 
Gerenuk said:
Using something which I think is called z-transform you get
S''''+(a+bx^2)S''+(c+dx^2)S=e_5x^5+\dotsb+e_0
I guess that's your initial problem in reverse :)
That's correct. In fact my equation is

S''''+(a+bx^2)S''+(c+dx^2)S=0,

with some inhomogenious boundary conditions.
 
Sorry for messing around and confusing people :)
But maybe the special case from that webpage helps a bit :)
 

Similar threads

Graduate Taylor expansion for a nonlinear system and Picard Iterations
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Is this Recursive Pattern in the Collatz Sequence Known?
  • · Replies 3 ·
Replies
3
Views
1K
Graduate Polar Fourier transform of derivatives
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Is the following sum a part of any known generalized function?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Fourier transform of a function in spherical coordinates
  • · Replies 10 ·
Replies
10
Views
6K
Undergrad Principal and Gaussian curvature of the FRW metric
  • · Replies 8 ·
Replies
8
Views
2K
Graduate Rearranging the equation for the cutoff condition in optical fibers
  • · Replies 2 ·
Replies
2
Views
1K
MHB Is the Uptake Equation Different When $k_2 = 0$?
  • · Replies 3 ·
Replies
3
Views
2K
Circuit for the inverse quantum Fourier transform
  • · Replies 2 ·
Replies
2
Views
6K
MHB Explore the Multiple Grid Method: Get Accurate Solution Faster
  • · Replies 36 ·
2
Replies
36
Views
8K