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May 21st, 2016, 02:58 PM   #1
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Help solving simple Recurrence Relation.

Hello,
I'm not a mathematician, but a physicist, so I have no idea how to solve those recurrence relations, but I need to solve one (it's related to a closure phase calculation in optical interferometry).

First of all I have the following recurrence relation:

$\displaystyle G(n \Delta x)=\phi[(n+1) \Delta x]-\phi (n \Delta x) - \phi(\Delta x)$

and it is given that a solution of this recurrence relation is:

$\displaystyle \phi\left ( n \Delta x \right ) = \sum_{k=1}^{n-1}G\left ( k \Delta x \right )+n \phi \left ( \Delta x \right )$

First of all, I believe I can represent it in a more simple way like:

$\displaystyle G(n)=\phi(n+1)-\phi (n) - \phi(1)$

with a given solution:

$\displaystyle \phi\left ( n \right ) = \sum_{k=1}^{n-1}G\left ( k \right )+n \phi \left ( 1 \right )$

So, first of all, as I said, I have no idea how to get from the given recurrence relation to the given solution. Can someone explain me the steps, please?

Secondly, I need to solve a slightly more complicated recurrence relation:

$\displaystyle G\left ( \sqrt{n^2+k^2} \Delta x\right) = \phi \left (\sqrt{\left ( n+1 \right )^2+k^2}\Delta x \right )-\phi\left ( \sqrt{n^2+k^2} \Delta x \right )-\phi \left ( \Delta x \right )$

which, again, I believe can be written as

$\displaystyle G\left ( \sqrt{n^2+k^2} \right) = \phi \left (\sqrt{\left ( n+1 \right )^2+k^2} \right )-\phi\left ( \sqrt{n^2+k^2} \right )-\phi \left ( 1 \right )$

I know, that for k=0, this relation reduces to the previous one, but I need a general solution for any k.
If the steps for solving the first one are the same, I think I will be able to solve it by myself, but I need your guidance on how to get to the given solution in the first example.

Thank you!



UPD: Forgot to mention, in case it's important:
$\displaystyle \phi(\Delta x)$ can be arbitrary (found from the physical measurements) and $\displaystyle \phi(0)=0$

Last edited by LmdL; May 21st, 2016 at 03:12 PM.
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May 22nd, 2016, 02:14 AM   #2
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Hey LmdL.

What you have described with the simple recurrence relation example (using G(k)) is a linear recurrence relation and these are found with the use of matrix techniques in terms of linear recurrence relations.

A quick google search found the document below:

http://nms.lu.lv/wp-content/uploads/...ecurrences.pdf

There are techniques (matrix wise) that depend on the number of terms and you should find this in any textbook/resource on linear recurrence relations.
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May 22nd, 2016, 02:51 AM   #3
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Chiro, thanks! I will try to use the steps described in your file.

UPD. I had a mistake in my original post. I need to solve the following relations, which depends both on n and k:

$\displaystyle G\left ( n,k \right) = \phi \left [\left ( n+1 \right ),k \right ]-\phi\left ( n,k \right )-\phi \left ( 1,0 \right )$
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May 22nd, 2016, 03:59 AM   #4
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I read the above file with steps on how to solve the recurrence relations. According to what they say, in my case it's a non-homogeneous recurrence, so I need to find a general solution to the homogeneous one and add a particular solution of a non-homogeneous one.

My recurrence:
$\displaystyle G(n )=\phi(n+1) -\phi (n) - \phi(1)$

Homogeneous part:

$\displaystyle \phi(n+1) -\phi (n) =0$

Characteristic equation:
$\displaystyle r^2 -r =0$

with roots:
$\displaystyle r=0,1$

So the general solution of a homogeneous one is:
$\displaystyle a_n=\alpha_1 0^n + \alpha_2 1^n = \alpha_2$

Now, I'm trying to guess a particular solution of a non-homogeneous one. In the file they say it should be similar to the non-homogeneous part of the relation, so my guess:
$\displaystyle b_n=c G(n) + d \phi (1) +f$

Substitution into non-homogeneous relation gives:
$\displaystyle G(n )=c G(n+1) + d \phi (1) +f -c G(n) - d \phi (1) -f - \phi(1)$
$\displaystyle G(n )=c G(n+1) -c G(n) - \phi(1)$
$\displaystyle c = \frac {G(n)+\phi(1)}{G(n+1) - G(n)} ; d = f =0 $

So, the final solution is:
$\displaystyle \phi (n) = a_n + b_n=\alpha_2 + \frac {G(n)+\phi(1)}{G(n+1) - G(n)} G(n)$

Where is my mistake?
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May 22nd, 2016, 05:01 AM   #5
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Ok, I found there my mistake was. I don't know how G(n) depends on n, so I cannot give a guess of b_n=cG(n)+d to find a particular solution of a non-homogeneous relation and need to solve it with power series guess.

Thanks, chiro!
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