matrix function like iteration without power series expansion andydude Long Time Fellow Posts: 510 Threads: 44 Joined: Aug 2007 07/09/2008, 02:38 PM Gottfried Wrote:Henryk's formula is (I inserted (x) at f°t) Hmm ... you mean Woon's formula ... andydude Long Time Fellow Posts: 510 Threads: 44 Joined: Aug 2007 07/09/2008, 02:57 PM I guess I should be more specific. The key observation to show the equivalence is that $n!\left({s \atop n}\right) = \prod_{k=0}^{n-1}(s-k)$ which means Woon's formula simplifies to: $ A^s = w^s \left( 1 + \sum_{n=1}^{\infty} (-1)^n \left({s \atop n}\right) \left[ 1 + \sum_{m=1}^{n} (-1)^m w^{-m} \left({n \atop m}\right) A^m \right] \right)$ and letting $w=1$ this simplifies to $ A^s = \left( 1 + \sum_{n=1}^{\infty} (-1)^n \left({s \atop n}\right) \left[ 1 + \sum_{m=1}^{n} (-1)^m \left({n \atop m}\right) A^m \right] \right)$ and assuming $0^0 = 1$ this simplifies to $ A^s = \left( \sum_{n=0}^{\infty} (-1)^n \left({s \atop n}\right) \left[ \sum_{m=0}^{n} (-1)^m \left({n \atop m}\right) A^m \right] \right)$ and combining (-1) terms, this simplifies to $ A^s = \sum_{n=0}^{\infty} \left({s \atop n}\right) \sum_{m=0}^{n} (-1)^{n+m} \left({n \atop m}\right) A^m$ and that is, as you call it, "Henryk's formula" from Woon's formula. Andrew Robbins bo198214 Administrator Posts: 1,616 Threads: 102 Joined: Aug 2007 07/09/2008, 06:18 PM andydude Wrote:Gottfried Wrote:Henryk's formula is (I inserted (x) at f°t) Hmm ... you mean Woon's formula ... You can also call it Newton's formula: $x^t = (x-1+1)^t = \sum_{n=0}^\infty \left(t\\n\right) (x-1)^n = \sum_{n=0}^\infty \left(t\\n\right)\sum_{k=0}^n \left(n\\k\right)(-1)^{n-k} x^k$ Woon just applied this to linear operators $A$ instead of $x$. This is possible because you dont need the commutativity for those formulas to stay true. However we cant calculate with iterations as with powers, because the composition is no more right distributive. $(f+g)\circ h=f\circ h+g\circ h$ but generally $f\circ (g+h)\neq f\circ g + f\circ h$. The binomial formula however relies on full (both side) distributivity. Especially generally $(f-\text{id})^{\circ n} \neq \sum_{k=0}^n \left(n\\k\right) (-1)^{n-k} f^{\circ k}$ The interesting thing however is that both expansions together are then again valid: $f^{\circ t} = \lambda^t \sum_{n=0}^\infty \left(t\\n\right) \sum_{k=0}^n \left(n\\k\right) (-1)^{n-k} \frac{f^{\circ k}}{\lambda^k}$ for each $\lambda$ as long as the right side converges. And I think this is new. Especially that the iteration by this formula is the same as regular iteration (which is sure for elliptic iteration, other cases are still to prove). Gottfried Ultimate Fellow Posts: 889 Threads: 130 Joined: Aug 2007 07/10/2008, 07:01 AM (This post was last modified: 07/10/2008, 07:32 AM by Gottfried.) I've got better convergence for the binomial (Woon-) method when computing f°0.5(x) by $\hspace{24} y_k = f^{\circ k+0.5}(x)$ (using Woon-method) $\hspace{24} f^{\circ 0.5}(x) = f^{\circ -k}(y)$ (using integer iteration) (but not yet for the Stirling-transformation. This became even worse) Also, the value, computed by diagonalization is verified (and seems to be the best approximation, see below) Here is the list of partial sums of the series for different k+0.5 (with a slight Euler-acceleration) Table of partial sums of series-terms for y_k = f°{k+0.5}(1), f(x)= sqrt(2)^x using 96 terms for all computations Code:k+0.5=  0.5    ...     5.5              6.5              7.5              8.5             9.5 ---------------------------------------------------------------------------------------------------   1.25000000000   1.25000000000   1.25000000000   1.25000000000   1.25000000000      1.25000000000   1.26110434560   4.49714780164   5.14435649285   5.79156518406   6.43877387527      7.08598256648   1.22525437136   1.93944375677  -3.72023614787  -5.88364612747  -8.42967369556     -11.3583188521   1.24668751623   5.99235259508   9.11068722540   13.5553576094   19.5849074109      27.4578802935   1.24076737182   1.49979923591  -5.23653818585  -11.6346957756  -21.7708420705     -36.9172871478   1.24335352318   4.28644727119   7.88117550221   15.1234185961   28.4432394460      51.2082273200   1.24296927970   0.47806011620  -2.27471644838  -8.89792678520  -23.1008859405     -50.9291782428   1.24330148390   2.68540223884   4.57545778450   9.77997787357   22.6057608416      51.2598956353   1.24334689776   1.52931366136  0.451609309727  -3.07391363987  -13.0838077583     -38.5219403266   1.24342527754   2.08739469898   2.71206223922   4.90465898282   11.8835163557      31.8389560094   1.24346627727   1.83531664757   1.56559795649  0.365479208323  -4.00667177875     -18.0519511170   1.24349944650   1.94309092062   2.11054106819   2.76424236321   5.31674044248      14.3664345470   1.24352286906   1.89908422835   1.86530727995   1.57378990702  0.216343075033     -5.15411825349   1.24354080559   1.91636602471   1.97064237771   2.13360031601   2.84254373454      5.84385041401   1.24355452136   1.90980233711   1.92717452801   1.88231915313   1.55966874840  -0.00183357885750   1.24356526803   1.91222417243   1.94449952880   1.99063473377   2.15814548109      2.95012751435   1.24357379457   1.91135286176   1.93780095156   1.94557320516   1.89002762276      1.52540063456   1.24358065583   1.91165944033   1.94032249372   1.96374161889   2.00591813819      2.18595121356   1.24358624145   1.91155367278   1.93939558496   1.95661723354   1.95739708093      1.89047738279   1.24359083767   1.91158952482   1.93972916723   1.95934236079   1.97714102763      2.01847035148   1.24359465614   1.91157756208   1.93961137808   1.95832298338   1.96931015585      1.96460639324   1.24359785628   1.91158149694   1.93965226197   1.95869669466   1.97234499833      1.98668941734   1.24360055956   1.91158021912   1.93963829064   1.95856216865   1.97119328258      1.97784819071   1.24360285978   1.91158062912   1.93964299792   1.95860979575   1.97162207811      1.98131215334   1.24360483013   1.91158049890   1.93964143237   1.95859318842   1.97146519999      1.97998157171   1.24360652831   1.91158053980   1.93964194690   1.95859889909   1.97152168166      1.98048347876   1.24360800024   1.91158052704   1.93964177964   1.95859696045   1.97150164438      1.98029729442   1.24360928281   1.91158053097   1.93964183347   1.95859761081   1.97150865646      1.98036530182   1.24361040586   1.91158052975   1.93964181631   1.95859739502   1.97150623337      1.98034081358   1.24361139372   1.91158053011   1.93964182174   1.95859746589   1.97150706093      1.98034951501   1.24361226636   1.91158052999   1.93964182004   1.95859744283   1.97150678136      1.98034646112   1.24361304032   1.91158053002   1.93964182057   1.95859745027   1.97150687485      1.98034752063   1.24361372931   1.91158053001   1.93964182041   1.95859744789   1.97150684388      1.98034715699   1.24361434483   1.91158053001   1.93964182046   1.95859744865   1.97150685405      1.98034728054   1.24361489654   1.91158053001   1.93964182044   1.95859744841   1.97150685074      1.98034723896   1.24361539259   1.91158053001   1.93964182045   1.95859744848   1.97150685181      1.98034725283   1.24361583993   1.91158053001   1.93964182044   1.95859744846   1.97150685147      1.98034724824   1.24361624445   1.91158053001   1.93964182045   1.95859744847   1.97150685158      1.98034724975   1.24361661124   1.91158053001   1.93964182045   1.95859744846   1.97150685154      1.98034724926   1.24361694464   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724942   1.24361724842   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724937   1.24361752584   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361777974   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361801259   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361822655   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361842353   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361860521   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361877306   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361892839   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361907236   1.91158053000   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361920601   1.91158053000   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361933026   1.91158053000   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361944593   1.91158053000   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361955376   1.91158053000   1.93964182045   1.95859744846   1.97150685155      1.98034724938   1.24361965442   1.91158053000   1.93964182045   1.95859744846   1.97150685155      1.98034724938 .... We see the improving of convergence for higher k. Here are the results of the second part of computation Table for f°0.5(1) computed by f°{-k}(y_k) Code:using   value for k+0.5   f°0.5(1) ------------------------------------------- _0.5:  1.24362081784741256884404697491 _1.5:  1.24362164528491826158633097226 _2.5:  1.24362162704002978026654871300 _3.5:  1.24362162769989241325230973243 _4.5:  1.24362162766649448252099408767 _5.5:  1.24362162766868372429248670409 _6.5:  1.24362162766850631390326437050 _7.5:  1.24362162766852353956309027978 _8.5:  1.24362162766852158056624384012 _9.5:  1.24362162766852183704563622465 10.5:  1.24362162766852179891157968560 11.5:  1.24362162766852180527979917972 12.5:  1.24362162766852180409621393150 13.5:  1.24362162766852180433916726086 14.5:  1.24362162766852180428444683747 15.5:  1.24362162766852180429789422049 16.5:  1.24362162766852180429430608135 17.5:  1.24362162766852180429534120427 18.5:  1.24362162766852180429501955967 19.5:  1.24362162766852180429512685495 20.5:  1.24362162766852180429508854399 21.5:  1.24362162766852180429510314755 22.5:  1.24362162766852180429509721882 23.5:  1.24362162766852180429509977687 Computed by Diagonalization method (with fixpoint-shift to fixpoint 2) Code:diag:  1.24362162766852180429509898361compare: Code:_0.5:  1.243620... _2.5:  1.2436216270... 18.5:  1.24362162766852180429502... 20.5:  1.24362162766852180429508... 22.5:  1.24362162766852180429509 | 721882... diag:  1.24362162766852180429509 | 898361... 23.5:  1.24362162766852180429509 | 977687... 21.5:  1.24362162766852180429510... 19.5:  1.24362162766852180429512... _1.5:  1.24362164... Gottfried Helms, Kassel andydude Long Time Fellow Posts: 510 Threads: 44 Joined: Aug 2007 07/14/2008, 06:21 PM bo198214 Wrote:And I think this is new. Especially that the iteration by this formula is the same as regular iteration (which is sure for elliptic iteration, other cases are still to prove). Hmm. Awhile back, I did some test with parabolic iteration, and convinced myself that Woon's formula and Jabotinsky's formula produce exactly the same results for parabolic iteration (for symbolic coefficients). So I could write up something about this if needed. Andrew Robbins bo198214 Administrator Posts: 1,616 Threads: 102 Joined: Aug 2007 07/14/2008, 09:55 PM andydude Wrote:Awhile back, I did some test with parabolic iteration, and convinced myself that Woon's formula and Jabotinsky's formula produce exactly the same results for parabolic iteration (for symbolic coefficients). So I could write up something about this if needed. The problematic thing about parabolic iteration is convergence. You usually only have an asymptotic development at the fixed point. I.e. the series is not developable at the fixed point however in every neighborhood and all the coefficients converge (towards the fixed point in a certain sector) to that of the asymptotic development. I also see that I made mistake in my first post about the convergence of that series. It is not true that the series always converges if the function has a finite attracting fixed point. « Next Oldest | Next Newest »

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