|
If I wanted to generate a sequence that follows the pattern $$x_{n}=x_{n-1}+\dfrac{1}{2}\left(\sqrt{1-y_{n-1}\ ^{2}}-x_{n-1}\right)\\ y_{n}=y_{n-1}+\dfrac{1}{2}\left(\sqrt{1-x_{n}\ ^{2}}-y_{n-1}\right)$$ rather than writing the whole thing out: ... etc. What is the best way to do it? |
||||
|
|
|
The previous answers correspond to the recursive model of the form $$[x_n,y_n]=[f(x_{n-1},y_{n-1}),g(x_{n-1},y_{n-1})]$$. However, the state-space model of the question is of the form
\begin{align}
[x_n,y_n]&=[f(x_{n-1},y_{n-1}),g(x_{n-1},y_{n-1},x_n)]\\
&=[f(x_{n-1},y_{n-1}),g(x_{n-1},y_{n-1},f(x_{n-1},y_{n-1}))]
\end{align}
The latter equation is the correct form that should be applied with the methods proposed by the previous answers. Another method could be to use
|
|||||
|
|
In an earlier edit, I had misread one of the subscripts in the y equation (thanks to Stelios for pointing this out). Here is the corrected version: To get many values: |
|||||
|
|
Credit goes to Stelios for spotting a mistake in the original answer :) |
|||||
|
|
Firstly,I simplify your formular: $$x_{n}=\frac{1}{2}\left(x_{n-1}+\sqrt{1-y_{n-1}\ ^{2}}\right)\\ y_{n}=\frac{1}{2}\left(y_{n-1}+\sqrt{1-x_{n}\ ^{2}}\right)$$ Seeing the sequences as below: $$x_1,y_1,x_2,y_2,x_3,y_3\ldots\\ $$ I define a function:$f(x,y)=1/2(x+\sqrt{1-y^2})$ $$ x_2=f(x_1,y_1)\\ y_2=f(y_1,x_2)\\ x_3=f(x_2,y_2)\ldots $$ So considering the underlying permutation : $$(x_1,y_1),(y_1,x_2),(x_2,y_2),(y_2,x_3) \\ \ldots$$ Solutions$x_0=1/2,y_0=\sqrt{3}/4$ To achieve:$x_0,y_0,x_1,y_1,\dots x_{12},y_{12}$
Lastly
|
|||
|
|
|
Using: Initial condition {0,0} first 10: yields: Note fixed points on $y=\sqrt{1-x^2}$:
|
|||||
|
