Poll: How do the different iterations relate?This poll is closed. regular iteration at both fixed points is equal and equal to the matrix power iteration 0% 0 0% Only the regular iteration at one fixed point is equal to the matrix power iteration 0% 0 0% all 3 iterations are different 100.00% 2 100.00% Total 2 vote(s) 100%
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 iteration of fractional linear functions bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 06/14/2009, 05:16 PM The interesting thing is that fractional linear functions, i.e. mappings of the form $ h_A(x)=\frac{ax+b}{cx+d}$ can be represented with help of the matrix $A=\begin{pmatrix}a &b\\c& d\end{pmatrix}$. as follows. The composition of these maps corresponds to the composition of their matrices! $h_{AB}=h_A \circ h_B$. We know this phenomenon from the Carleman matrix! But $A$ is *not* the Carleman matrix of $h_A$. The representation is only unique up to a fraction extension constant: $h_A = \operatorname{id}\Leftrightarrow A = \alpha I$ for some complex $\alpha$. Thatswhy we have here a natural way of fractionally iterating these fractional linear functions, i.e. via matrix powers. I vaguely remember that Gottfried posted somewhen a link to a thread on sci.math that also discusses the iteration of fractional linear functions (so if you still know, Gottfried, perhaps you can repost it here). Further investigation shows that indeed the eigenvalues of the matrix are non-real if and only if the function has non-real fixed points. As we know the fractional iteration via matrix powers is given by linear combinations of powers of the eigenvalues of the matrix. So it seems that the iteration via matrix powers is linked to the iteration at the two fixed points (which is also real if the fixed point is real). So now my question to the audience of this forum. We have only one matrix power iteration but we have the regular iteration iteration at the two fixed points of $h_A$. So the question is how do they relate. I make a poll and ask you about your opinions (I was just too lazy yet to verify it theoretically which should not be too difficult, but at least that way we can check how good our intuition is.) tommy1729 Ultimate Fellow Posts: 1,493 Threads: 356 Joined: Feb 2009 06/14/2009, 11:33 PM (06/14/2009, 05:16 PM)bo198214 Wrote: The interesting thing is that fractional linear functions, i.e. mappings of the form $ h_A(x)=\frac{ax+b}{cx+d}$ can be represented with help of the matrix $A=\begin{pmatrix}a &b\\c& d\end{pmatrix}$. as follows. The composition of these maps corresponds to the composition of their matrices! $h_{AB}=h_A \circ h_B$. We know this phenomenon from the Carleman matrix! But $A$ is *not* the Carleman matrix of $h_A$. The representation is only unique up to a fraction extension constant: $h_A = \operatorname{id}\Leftrightarrow A = \alpha I$ for some complex $\alpha$. Thatswhy we have here a natural way of fractionally iterating these fractional linear functions, i.e. via matrix powers. I vaguely remember that Gottfried posted somewhen a link to a thread on sci.math that also discusses the iteration of fractional linear functions (so if you still know, Gottfried, perhaps you can repost it here). Further investigation shows that indeed the eigenvalues of the matrix are non-real if and only if the function has non-real fixed points. As we know the fractional iteration via matrix powers is given by linear combinations of powers of the eigenvalues of the matrix. So it seems that the iteration via matrix powers is linked to the iteration at the two fixed points (which is also real if the fixed point is real). So now my question to the audience of this forum. We have only one matrix power iteration but we have the regular iteration iteration at the two fixed points of $h_A$. So the question is how do they relate. I make a poll and ask you about your opinions (I was just too lazy yet to verify it theoretically which should not be too difficult, but at least that way we can check how good our intuition is.) actually i use ordinary algebra for this ... no fancy stuff :p unless you want a superfunction or such. for half - iterate , just ordinary algebra. besides what about the case of a single fixed point ? bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 06/16/2009, 02:05 PM Unfortunately someone else was already faster. There is a very recent (May 2009) Chinese article: Shi, YongGuo; Chen, Li: Meromorphic iterative roots of linear fractional functions,Science in China Series A: Mathematics Vol 52 Iss 5, p. 941-948 If you are interested, download quickly; I dont know how long this link will be available for free. (06/14/2009, 11:33 PM)tommy1729 Wrote: actually i use ordinary algebra for this ... ... for half - iterate , just ordinary algebra. Well then demonstrate. tommy1729 Ultimate Fellow Posts: 1,493 Threads: 356 Joined: Feb 2009 06/16/2009, 10:18 PM (This post was last modified: 06/16/2009, 10:31 PM by tommy1729.) (06/16/2009, 02:05 PM)bo198214 Wrote: Unfortunately someone else was already faster. There is a very recent (May 2009) Chinese article: Shi, YongGuo; Chen, Li: Meromorphic iterative roots of linear fractional functions,Science in China Series A: Mathematics Vol 52 Iss 5, p. 941-948 If you are interested, download quickly; I dont know how long this link will be available for free. well actually many such papers are written regularly. also for free ; i remember han de bruijn posting one on sci.math. i " wrote " about some too , however " wrote " is literal : on paper in your pdf , you find an answer to a question , you asked me recently. more particular , i said polynomials of degree 2 have no iterate root from C to C. you asked " why " and " are you sure " well ref 12 in the pdf is why i quote from the first (!) page , the introduction (!) : " Concerning polynomial functions, the result of [12] implies that quadratic polynomial functions have no n-th (n >= 2) iterative root from C to itself. Similar results concerning some other polynomials can be found in [13]. " which thus confirms what i said , already in the introduction. high regards tommy1729 (06/16/2009, 02:05 PM)bo198214 Wrote: (06/14/2009, 11:33 PM)tommy1729 Wrote: actually i use ordinary algebra for this ... ... for half - iterate , just ordinary algebra. Well then demonstrate. simple. just use f(x) = a' x + b' / c' x + d' and expand f(f(x)) now set f(f(x)) equal to your a x + b / c x + d and solve for a' b' c' and d' then f(x) is your half-iterate by ordinary algebra. trivial. regards tommy1729 Gottfried Ultimate Fellow Posts: 789 Threads: 121 Joined: Aug 2007 06/17/2009, 07:50 AM (06/14/2009, 05:16 PM)bo198214 Wrote: The interesting thing is that fractional linear functions, i.e. mappings of the form $ h_A(x)=\frac{ax+b}{cx+d}$ can be represented with help of the matrix $A=\begin{pmatrix}a &b\\c& d\end{pmatrix}$. as follows. The composition of these maps corresponds to the composition of their matrices! $h_{AB}=h_A \circ h_B$. We know this phenomenon from the Carleman matrix! But $A$ is *not* the Carleman matrix of $h_A$. The representation is only unique up to a fraction extension constant: $h_A = \operatorname{id}\Leftrightarrow A = \alpha I$ for some complex $\alpha$. Thatswhy we have here a natural way of fractionally iterating these fractional linear functions, i.e. via matrix powers.Hi Henryk - quite interesting; though it didn't come to my attention when I fiddled a bit with functions like this. Quote:I vaguely remember that Gottfried posted somewhen a link to a thread on sci.math that also discusses the iteration of fractional linear functions (so if you still know, Gottfried, perhaps you can repost it here). I'm sorry, I have a vague idea of such postings but no true memory. Perhaps if we look for posts of alain verghote, or we may email him personally. He seems to have experiences with/lists of several of such type of functions. Alain Verghote alainverghote (att) gmail(.)com I'm not much with math this days, so better someone else gets in contact with him. He's a friendly person who likes to exchange about that subject. Gottfried Gottfried Helms, Kassel bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 06/17/2009, 09:46 AM (06/16/2009, 10:18 PM)tommy1729 Wrote: well actually many such papers are written regularly.regularly maybe a bit of exaggeration, isnt it? How can one find news about a restricted topic on a regular basis? Quote:in your pdf , you find an answer to a question , you asked me recently. more particular , i said polynomials of degree 2 have no iterate root from C to C. you asked " why " and " are you sure " Help my memory, where did you ask that? On the other hand its not that surprising that they have no entire iterative root. Similar to that it has no entire arithmetic root. But as soon as we remove a suitable cut from the complex plane, it has iterative as well as arithmetic roots. Quote:well ref 12 in the pdf is why I already posted that reference two months ago on this forum here. (06/16/2009, 02:05 PM)bo198214 Wrote: simple. ... trivial. Well I guess from many problems one can pick easy particular cases. Actually fractional iteration refers to a formula for any (fractional) iteration count, not just 1/2. As long as you restrict yourself to simple cases, you may not need advanced methods. bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 06/17/2009, 09:47 AM (This post was last modified: 06/17/2009, 09:52 AM by bo198214.) (06/17/2009, 07:50 AM)Gottfried Wrote: quite interesting; though it didn't come to my attention when I fiddled a bit with functions like this.I found it again. Its your post about iteration of 1/(x+1). You dont mention the sci.math thread explicitely, but I think I had a look it at it at that time and thatswhy remembering that some people had quite a knowledge about iteration of fractional linear maps. May be I read Alain there. You asked in your thread about different methods of iterating 1/(x+1). This is a particular case of iterating a fractional linear map. Kinda this thread is a generalization of your problem. I see the 3 methods: regular iteration at the first fixed point, regular iteration at the second fixed point, and fractional iteration of the characteristic 2x2 matrix. In your web-page you do regular iteration at one fixed point (via your matrix method). But it is not clear to me whether you compare the result with the regular iteration at the other fixed point. « Next Oldest | Next Newest »

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