matrix function like iteration without power series expansion  Printable Version + Tetration Forum (https://math.eretrandre.org/tetrationforum) + Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) + Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) + Thread: matrix function like iteration without power series expansion (/showthread.php?tid=186) Pages:
1
2

matrix function like iteration without power series expansion  bo198214  06/30/2008 Guys! That I didnt see that before! We have a very simple formula for computing the th iterate of an arbitrary function: This series does not always converge but at least if has an attracting fixed point reachable from then converges, because then is bounded, say , then . I.e. the sum is absolutely convergent. I bet my pants that this is the regular iteration at the lower (attracting) fixed point in the case , . Why is this matrix function like? Well, if has eigenvalues all smaller than 1 then, we dont need the Jordanization of , but the matrix power is also given as a convergent series . If is the Carleman matrix of , then we just consider the first row in all those matrix powers and add them up. Dont make the error however to assume that This is not true. You can only do this with matrices as while mostly . Notes:
RE: matrix function like iteration without power series expansion  Gottfried  06/30/2008 Using a matrixexpression this would be t°h(x) = W^1 sum k=0..inf sum j=0..k (1)^j * binomial(k,j) *diag(1,u^j,u^2j,...) * W sum j=0..k (1)^j * binomial(k,j) *dV(u^j) = diag(u^j1) *PPow( sum k=0..inf bo198214 Wrote: Hmm, let me try (maybe I didn't get this right yet). is which is just a binomial weighting of the coefficients c_n. In my analyses I got the coefficients so, for instance, the c_n for the halfiteration are The rate of growth of the a_ncoefficients for t=0.5 was asymptotically where m_n are also growing coefficients, if only the leading coefficient of the polynomials at x^n are taken into account. Now the quotient of two consecutive binomials seem to approach 1, so the strong growth of about u^n^2/n!, or the quotient of two consecutive coefficients of ~ u^2n/n seems to dominate the characteristic of the c_ncoefficients. A series with quotient of increasing absolute value u^(2n)/n, u>1 cannot regularly be Eulersummed; maybe it can be summed with Borelsummation of higher orders. To be "not regularly" Eulersummable does not mean, we cannot have an approximation of a certain degree; however the problem with this is, that the partial sums may converge up to a certain index n, from where it "begins to diverge"  and it is not yet known to me, to what extent we can use the intermediate approximated value  I'm investigating for verification of some experimental summationmethods of the required power. Hmm  i hope this is not more confusing than clarifying  I've my head not really free today (have to prepare the final lesson tomorrow) Gottfried RE: matrix function like iteration without power series expansion  bo198214  07/01/2008 Gottfried Wrote:Using a matrixexpression this would bewhat? Quote:"not regularly" Eulersummable Yes, I see. Its somehow similar to your tetra series its only summable via the matrix method for not having an attracting (finite) fixed point, i.e. in our case . However the aim was to not use the matrix method for computation. This is faster and does not require the function to be developable into a powerseries. For regular iteration we have both: limit formulas and powerseries coefficient formulas. And now we have also both for the matrix function method (if it can still be called that way for the limit formula). The limit formula yields exactly the same iterates as the matrix function method if it converges. With this formula its perhaps easier to verify that the matrix function method just gives the regular iteration at the attracting fixed point, though I didnt try to prove this yet. RE: matrix function like iteration without power series expansion  Gottfried  07/01/2008 bo198214 Wrote:Gottfried Wrote:Using a matrixexpression this would bewhat? ...editingcrap. Quote:Quote:"not regularly" Eulersummable Yes , sure  I only introduced the matrixmethod to have an idea about the estimated coefficients. I'm testing a summationmethod, which allows to sum series of such rate of divergence. There are these Hausdorffmeans, tightly related to the Rieszmethod. Unfortunately it is not obvious, how  for a certain selection of parameters  the range of applicability can be determined. I'm comparing the results of my parametersettings with known results for strongly diverging series and also I'm looking for articles on this... Gottfried z.B.: Über Klassen von Limitierungsverfahren, die die Klasse der Hausdorffschen Verfahren als Spezialfall enthalten Endl, Kurt in: Mathematische Zeitschrift, volume: 65 pp. 113  132 Online at digicenter Göttingen RE: matrix function like iteration without power series expansion  Gottfried  07/02/2008 bo198214 Wrote:Guys! That I didnt see that before! I want to add, that this expression exites me very much. Just these days I was looking for alternatives (hopefully improvements) of our fractional iteration problem. I was looking for two aspects
The first is obvious; the idea is that only values are involved, which are provided by the function itself (so, for instance, possibly only of a relevant subset of complex numbers which are accessible by the function and its iterates itself/themselves  hmm, amateurish expressed; I hope the idea behind this can be understood anyway) The second, because the fractional binomials use gamma of fractional parameters. So  in my view  this is a big shot... Gottfried RE: matrix function like iteration without power series expansion  Gottfried  07/08/2008 Hmm, rereading this all, it appears to me, that I might not have understood Henryk's query correctly.[update:removed silly question] Gottfried RE: matrix function like iteration without power series expansion  andydude  07/08/2008 This seems to be S.C.Woon's method. If you take Woon's doublebinomial expansion and let w=1, then you should get the formula you gave. I wonder if it is equivalent, or slightly different? Andrew Robbins RE: matrix function like iteration without power series expansion  Gottfried  07/08/2008 Henryk  I crosschecked some results of your method against the diagonalizationmethod. I got some differences, which look somehow suspicious to me. Using, for instance, u = 0.9 (to be near at a critical value) t = exp(u) = 2.45960311116 b = t^(1/t) = 1.44182935647 I got y = b°0.5(1) = 1.25590667... where in these digits no difference appear. But I got differences in less significant digits, which as I said look suspicious: either my summationmethods are not perfect (in fact they give different approximations  I'm still searching for better ones), or I need simply more terms (opposite to the impression). They stabilize (using 96 terms) with differences in the last partial sums of order 1e18 to 1e22. So they seem to give usable results. However, the difference to your binomialmethod y<binomial>  y<matrix> ~ 0.00000000259698761292 is so large compared to the summationinternal inaccurcy, that this seems to be systematic. Different summationmethods gave different result even worse  so in any case I've to improve my summation for these type of divergent series. For t=2, b=sqrt(2) it was much better, but possibly things were simply not visible... I hope I can locate (and possibly remove) the source of the difference... Gottfried RE: matrix function like iteration without power series expansion  andydude  07/08/2008 andydude Wrote:This seems to be S.C.Woon's method. Just for reference, here is the article. The general formula is given on page 32 (formula 71), and I believe it is the same as the one given above. Andrew Robbins RE: matrix function like iteration without power series expansion  Gottfried  07/09/2008 While I played around with some transformations to find a meaningful transformation for a summation in the divergent case, I just found a transformation of the problem, which at least provides an improvement for the approximation for the convergent case. Henryk's formula is (I inserted (x) at f°t) Here I rewrite this in vectorial notation, for simpliness. With this Henryk's formula may be rewritten as product of vectors V1 and V2 where ("col" means here: a columnvector of index/length as subscripted) into where Here S1(r,c) means the Stirlingnumber 1st kind of row=r,col=c Using f(x) = b^x, f°t(x) = f°(t1)(b^x) , with b=sqrt(2) and t=0.5 I get different series with the two methods. Here is the comparison (the numbers are the individual terms from the vectorproduct V1~*V2): The problem for the approximation of a useful result based on these methods is the (slow) monotone decrease (or increase) of these series and hence of the partial sums. That may also be the reason for the observed difference of the diagonalizationmethod to this type of computation: even using 200 and more terms (as I did) leaves a remainder in the order of 1e10 or the like. The nice thing is now, that we have an interval for the result, from where we may interpolate a better estimate. However, I don't see it yet (to use the mean seems to be too simple...) 