# Iterative Equations in Wolfram Alpha

I am trying to solve an iterated equation, but it is quite messy and I am having trouble. Here are my equations:

$$\mu_{t+1} = \frac{h_t \mu_t + h_\epsilon Z_t}{h_t + h_\epsilon}$$

and

$$h_{t+1} = \frac{(h_t + h_\epsilon) h_\delta}{(h_t + h_\epsilon + h_\delta)}$$

$$h_0$$, $$m_0$$, $$h_\epsilon$$, $$h_\delta$$ and $$Z_t$$ are all given. I would like to solve for $$m_t$$ in terms of the givens.

Is it possible to solve this equation in Wolfram Alpha? If not, is there a different software I could use to solve it? If both those fail, does anyone have ideas as to how I might solve it by hand? I've been struggling with it for some time, but it gets very messy very quickly!

• I see no reason to think there is a closed form. Commented Feb 27, 2020 at 18:17
• @GEdgar, what do you mean? Shouldn't it be possible to express in terms of sums or products of the initial parameters and the $Z_t$ terms? Commented Feb 27, 2020 at 18:24
• @GEdgar Yes there is one : see my answer. Commented Feb 27, 2020 at 19:32
• @GEdgar It's extraordinary : I just saw your answer : we were on the same tracks ! Commented Feb 27, 2020 at 19:34
• Thanks all for your help! Commented Feb 28, 2020 at 21:26

I think this will be useless...
The second equation does not involve $$\mu_t$$ at all, so we solve only for $$h_t$$. We must find the $$t$$ power of a certain $$2 \times 2$$ matrix.
Result (from Maple): $$h_t = \frac{A_t h_0+ B_t}{C_t h_0 + D_t}$$ Where $$A_t = \left( \sqrt {h_\epsilon\, \left( 4\,h_\delta+h_\epsilon \right) }h_0+h_\epsilon\, \left( h_0-2\,h_\delta \right) \right) \left( { \frac {2\,h_\delta+h_\epsilon-\sqrt {h_\epsilon\, \left( 4\,h_\delta+h_\epsilon \right) }}{h_\delta}} \right) ^{t}\\ B_t = \left( \sqrt {h_\epsilon\, \left( 4\,h_\delta+h_\epsilon \right) }h_0-h_\epsilon\, \left( h_0-2\,h_\delta \right) \right) \left( {\frac {2\,h_\delta+h_\epsilon+ \sqrt {h_\epsilon\, \left( 4\,h_\delta+h_\epsilon \right) }}{h_\delta}} \right) ^{t} \\ C_t = \left( \sqrt {h_\epsilon\, \left( 4\,h_\delta+h_\epsilon \right) }-2\,h_0-h_\epsilon \right) \left( {\frac {2\,h_\delta+h_\epsilon-\sqrt {h_\epsilon\, \left( 4\,h_\delta+h_\epsilon \right) }}{h_\delta}} \right) ^{ t} \\D_t= \left( \sqrt {h_\epsilon\, \left( 4\,h_\delta+h_\epsilon \right)}+2h_0+h_\epsilon \right) \left( {\frac {2\,h_\delta+h_\epsilon+\sqrt {h_\epsilon\, \left( 4\,h_\delta+h_\epsilon \right) }}{h_\delta}} \right) ^{t}$$

Your second equation is of the form :

$$h_{n+1}=\dfrac{ah_n+b}{ch_n+d}\tag{1}$$

Therefore there is an explicit solution (see below in (2)) of $$h_n$$ as a fonction of $$h_0$$ and $$n$$.

Having this expression, it remains to plug it into the first equation.

Explanation for the explicit solution :

There is a formula one can find as "rational difference equation" in (https://en.wikipedia.org/wiki/Iterated_function) :

$$h_n=\frac{a}{c}+\frac {bc-ad}{c}\left[\dfrac{A\alpha^{n-1}-B\beta^{n-1}}{A\alpha^{n}-B\beta^{n}}\right]\tag{2}$$

with

$$\alpha =\dfrac {a+d+{\sqrt {(a-d)^{2}+4bc}}}{2} \ \text{and} \ \beta =\dfrac {a+d-{\sqrt {(a-d)^{2}+4bc}}}{2}$$

$$A=c \; h_0-a+\alpha \ \ \ \text{and} \ \ \ B=c \; h_0-a+\beta$$

(I thought yesterday that there is a sign error in the numerator ; but this is not the case).

Remarks :

1) Important : the formulas assume that $$\Delta=(a-d)^2+4bc \geq 0$$ which is the case for you ($$\Delta=h_{\delta}^2>0$$). Usually, sequence $$h_n$$ converges to a fixed value which is one of the roots of quadratic "fixed point" equation $$H=(aH+b)/(cH+d).$$

2) one can give a nice linear algebra explanation for (2) where $$\alpha$$ and $$\beta$$ upcome as eigenvalues of matrix $$\begin{pmatrix} a&b\\c&d \end{pmatrix}$$.

• Can you confirm that, "physically" (I imagine that this issue is connected to a physical evolution as a function of dicretized time) $h_t$ is bound to be tending to a finite limit ? Commented Feb 28, 2020 at 17:41
• Hi miss, do you know how to derive the above equation you got from wikipedia? Commented Jul 9, 2021 at 13:04
• You can post your answer here at math.stackexchange.com/questions/4194009/… Commented Jul 9, 2021 at 13:05
• I have read the article that is Link 20 from wiki, but the result is not the same as the wikipedia one Commented Jul 9, 2021 at 13:06