A nice series for b^^h , base sqrt(2), by diagonalization  Printable Version + Tetration Forum (https://math.eretrandre.org/tetrationforum) + Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) + Forum: Computation (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=8) + Thread: A nice series for b^^h , base sqrt(2), by diagonalization (/showthread.php?tid=249) Pages:
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RE: Logarithmic behaviour of the super exponential at 2  bo198214  03/11/2009 Gottfried, we discussed that already in an much earlier post. Your whole infinite matrix computation is just r e g u l a r i t e r a t i o n !!! Let me make it clear again: Gottfried Wrote:If we have the Bellmatrix B for b^x , then we cannot invert B for the infinite case. But First Bell matrix behave antiisomorphic: let B[f] be the Bell matrix of f at 0, then B[fog]=B[g]*B[f] Now let us assign the variables: B is the Bellmatrix of at 0. P~ is the Bellmatrix of fS2F is the Bellmatrix of (for my convenience I omit the base sub script at exp and dexp) As dexp has a fixed point at 0 fS2F is upper triangular. Your equation just corresponds to Quote:But P (as well as then PInv) are the binomialmatrices and they perform addition when operating on the powerseries: exactly: , the inverse of is . Quote:Then rearranging the invertible P as PInv to the left Thats clear without touching any matrix: V(e^x  1) is the Bell matrix of dexp, and fS1F is the inverse of the Bell matrix of dexp: Quote:I do the same with the eigensystemdecomposition / Schröderfunction. I found the fixpointshift with my matrixnotation by simply proceeding from the initial equation C is the Bell matrix of texp. we see that texp has 0 as fixed point and hence is the Bell matrix triangular. And the power can be taken exactly. Quote:All in all I use the "regular iteration with fixpointshift", but only as far as I can represent it coherently in terms of infinite matrices/known closedform expressions for sums of infinite powerseries, which result by the implicte dotproducts. But I still dont see what it brings to rewrite an established method with matrices. Whether you compute it with matrices, with direct powerseries formulas, or with limit formulas (without powerseries), whether with Abel or super function, the result is always the same, its always regular iteration. Quote:Thus I have the difficulties with b>eta, since the occuring complexvalued matrices C give unsatisfying powerseries, and I have not yet the remedy to deal with that series appropriately. I gave the answer already in the previous post. The resulting power series (where are the coefficients obtained from ) has convergence radius t in your notation. It converges at most for on the real axis. Quote:Yes, thanks! It's progressing diabetes, and sometimes I'm fitter, sometimes struck down, and in general just less powerful and regenerable than up to recent years. Just life.. For me it seems you should do something healthy not always brood about the same things. Spring is coming, go out, have a look at the blue sky, or at the young girls passing RE: Logarithmic behaviour of the super exponential at 2  Gottfried  03/11/2009 Hi Henryk  just a short reply here. bo198214 Wrote:Gottfried, we discussed that already in an much earlier post. Well, I felt it is needed to make it explicitely, that it is special because of the (even blueish enhanced) hyperlink: Quote:The matrix power approach makes use of the established method to obtain noninteger powers (and other analytic functions) of finite matrices via diagonalization. where the keyword "finite matrices" occur. We had that several times and since I had assumed, that I had made it clear that I'm always working on infinite matrices, that remark is surprising. Well, next question. "Why work with matrices if things are otherwise well known..."  I still don't claim something special. It's just my path into the matter: I came from a project, in which I compiled relations between pascal Stirling, Euler and other matrices operating on formal powerseries and stumbled on the possibility of iterating functions (other than that of addition, which I had already studied to some nice encounters with the zeta/etafunction) Here I had the iteration of the exponentialfunction and as you might (frustrated? ) remember a hard and long time to even explain what I'm doing although Aldrovandi, Woon, Comtet and others were long time around in the scene. So its for personal reasons (experience with my matrix "toolbox"), possibly for "historical reasons" (to keep a track consistent) and recently for the connection of iterationseries with series of matrixpowers, which I've not seen yet except possibly in the form of the umbralcalculus, where it might be hidden behind the scene. Just recently someone in sci.math asked for a set of symbolic matrix operations for mathematica or maple. Could be a nice start... I've just reread Andrew's "exact entries for slogoperator" today and found a similar matrixdiscussion there: it helps to understand since I've no training in functional analysis, the bit I had was in 1972 to 77 and only in relation for the computercourses, which were my main subject.  Well, but let's not lose the track here, for which I opended the thread. I think I'll have a much closer look at that series for tetration next time, to see whether and if, how, it's a special one.  And, yes: Quote:For me it seems you should do something healthy not always brood about the same things. Spring is coming, go out, have a look at the blue sky, or at the young girls passing that's certainly what I should do. Hope I'll get things working/walking. And why the heck should the girls *pass*? Ok  wish you all a good night Gottfried RE: Logarithmic behaviour of the super exponential at 2  bo198214  03/11/2009 Gottfried Wrote:Well, I felt it is needed to make it explicitely, that it is special because of the (even blueish enhanced) hyperlink: For me matrix power method means exactly what I wrote. Take the truncations of an infinite matrix , apply the matrix power and take the limit . Now one can apply this method at different development points, i.e. the original function is conjugated to the development point : or written as composition: then we define the application of the matrix power method at point p by . This is the general method. If I now apply the matrix power method to a fixed point p, then the Bell/Carlemann matrix of is lower/upper triangular. For those matrices the power of the truncation is the same as the truncation of the power. That means the limit to infinity is just an expansion of the matrix, once computed values do not change in that process. So we see that regular iteration is the particular case of applying the matrix power method to a fixed point. And I really honored this method (applied to a *nonfixed point* like 0) because it can do where regular iteration fails: to be able to compute real iterates for . This also puzzles me that despite you insinst on regular iteration for . Quote:Well, next question. "Why work with matrices if things are otherwise well known..."  I still don't claim something special. It's just my path into the matter: I came from a project, in which I compiled relations between pascal Stirling, Euler and other matrices operating on formal powerseries and stumbled on the possibility of iterating functions (other than that of addition, which I had already studied to some nice encounters with the zeta/etafunction) But if you focus too much on only matrices and nothing else, such interesting relations like the convergence radius of the iterates is just out of scope for you. Because it is derived by the interelation of the limit formulas for regular iteration (which is power series free) and power series formulas for regular iteration. Quote:I've just reread Andrew's "exact entries for slogoperator" today and found a similar matrixdiscussion there: it helps to understand since I've no training in functional analysis, the bit I had was in 1972 to 77 and only in relation for the computercourses, which were my main subject. Wah I was born in that time Quote:that's certainly what I should do. Hope I'll get things working/walking. And why the heck should the girls *pass*? Well you are right, they build up a big cluster around your place. RE: Logarithmic behaviour of the super exponential at 2  Gottfried  03/14/2009 bo198214 Wrote:For me matrix power method means exactly what I wrote. Take the truncations of an infinite matrix , apply the matrix power and take the limit The difference for the matrixpower may be "small"; but it may be significant when applied to the matrixinverse: "takethetruncated,invert,useasapproximation" may give then different results from "concludetheexactentriesforinfinitecase,truncate,useasapproximation". It is even more significant, if we discuss more complex entities like the set of eigenvalues. So these different views of things should be still explicite, and it would be good to keep identifying nomenclature. I tended to give the finitematrixbased the attribute "polynomially", but this might be not the best choice... Quote:Now one can apply this method at different development points, i.e. the original function is conjugated to the development point :Surely. No dissent here. Quote:And I really honored this method (applied to a *nonfixed point* like 0) because it can do where regular iteration fails: to be able to compute real iterates for . This also puzzles me that despite you insinst on regular iteration for .:) As it comes to honor... Well, that's not my problem. I surely should have come to my full descriptiontext about my way of thinking in a new pdffile  I've some first chapters, but it's very complex and I stuck several times soon. I'll be "honoring" that method, too, so we have also no dissent here. I've just left this field and am digging at the other one for the gold. I think if a definitive description for the matrices in the infinite case can be given, (based on the hypothese about eigenvalues) this would be very good and if then also a method for the actual computation were found  this would please me much more than the approximating of 7^^Pi using adhoceigensystems of finitesizematrices. Maybe the latter will even be the only way to get to practical values; but then: well, there'll be many people, programs which could do that, very fine, why should I bother, it's not my job/profession/money to calculate values? Quote:But if you focus too much on only matrices and nothing else, such interesting relations like the convergence radius of the iterates is just out of scope for you. Because it is derived by the interelation of the limit formulas for regular iteration (which is power series free) and power series formulas for regular iteration.Here you made a point. However, not in the sense of missing the aspect of convergenceradius; in the contrary: I think I need the matrixlayout for the infinite case to have even better conditions for convergence considerations. And since in important cases we'll miss convergence anyway we can check for summability methods to overstep the range of converge, but in a wellfounded manner. But as I learned in some discussions in sci.math in the last monthes it is fruitful to discuss iteration also in terms of the functions themselves  even some very nice and surprising closed forms were discussed which I never could have found with the formal powerseries/matrixapproach. Here opened a much interesting field and I'm actually fiddling with that on a casual manner (you've notices my casual "iteration exercises" also here in the forum). I think I'll go into this much more if I have the feeling, that my questions/ideas with the infinite matrices are solved (or shown to be unsolvable) and I can close that case. Just currently I've applied the (infinite) matrixconcept to Andrew's slog with a nice achievement of insight... :) So there's still something in it. (I'll need it also for the discussion of iteration series, I think) RE: Logarithmic behaviour of the super exponential at 2  andydude  03/31/2009 bo198214 Wrote:then the limit can be given as You know, at first I was confused, because this formula implies that but the Cz page gives but then I realized the point was different Andrew Robbins RE: A nice series for b^^h , base sqrt(2), by diagonalization  andydude  04/06/2009 Gottfried Wrote:Here for base b = sqrt(2) : I used regular iteration, and I did not get these expansions at all. I got so what this series represents would be regular iteration, evaluated at , and expanded about . Is that right? How did you get this? Andrew Robbins RE: A nice series for b^^h , base sqrt(2), by diagonalization  Gottfried  06/10/2009 (04/06/2009, 10:23 PM)andydude Wrote:Gottfried Wrote:Here for base b = sqrt(2) : Hi Andrew  sorry, took a long time to answer. The coefficients occur as sums; your formula contains the xparameter, maybe if you insert x=1 we get identity. [update] I've the same coefficients as yours, just evaluated at x=1;see next post [/update] Let me explain in my matrix/PariGPnotation how I got the coefficients: Assume the notation of variables as usual: the variables b,t,u encoding the baseparameters, where for our example t=2, u=log(t) , b=t^(1/t) = sqrt(2). Also let exp_b°h(x) the h'th iteration of expfunction to base b, and dxp_t°h(x) the h'th iteration of the decremented expfunction to base t where we use the identity (if we have a real fixpoint) which will be numerically with x=1, t=2 The function dxp to base t (=2) gets its coefficients by the triangular Bellmatrix Ut, where we implement the h'th fractional power using diagonalization, denoting the eigenmatrices as W and WInv (=W^1) Code: ´ Ut^h = W * dV(u^h) * WInv and the function is in general finally computed by Code: ´ V(y)~ = V(x/21)~ * Ut^h Keeping the iterationparameter variable we have, using the eigenmatrices Code: ´ V(y)~ = V(x/21)~ * W * dV(u^h) * WInv Since we assume x being constant x=1, we can precompute the rowvector S~ Code: ´ S~ = V(1/2)~ *W which implements also a Schröderfunction (need not be the principal one) for dxp_t°h(x) at the constant x=1/2 . This is also the schröderfunction for exp_b°h(x) at x=1, as fas as we look only on the coefficients of the 2'nd column of W. So we can write Code: ´ V(y)~ = S~ * dV(u^h) * WInv WInv provides the inverse of the schröderfunction, and if we extract only the scalar result we can write ( with the notation W[,1], meaning the second column of a marix W ) Code: ´ y = S~ * dV(u^h) * WInv [,1] Here we can interchange the the order of multiplication of the last two factors and precompute the constant coefficients of a powerseries in the resultvector M of Code: ´ M~ = S~ * diag(WInv[,1]) and get and according to the above Code: ´ exp_b°h (1) = 2 + 2*y we have the source of my coefficients in the vector M: Code: ´ exp_b°h (1) = 2 + 2*y = 2 + 2* M~ * V(u^h) So the coefficients, which I provided are just the precomputed coefficients in M, which represent the evaluation of the coefficients of the Schröderfunctions for dxp, including a fixpointshift of the xparameter and of the functionvalue.  In short, omitting matrices: Let C(x) denote a schröderfunction for dxp_t(x), c_k the k'th coefficient of its powerseries, D(x) the inverse and d_k the k'th coefficient of its powerseries, for brevity C and D the functions at x=1/2 and v = u^h, the h'th power of u. Then we can write and the k'th coefficient in my first mail is just 2* C^k * d_k in the formula above. Gottfried RE: A nice series for b^^h , base sqrt(2), by diagonalization  Gottfried  06/10/2009 (04/06/2009, 10:23 PM)andydude Wrote:Gottfried Wrote:Here for base b = sqrt(2) : Hi Andrew, 2'nd note. I just looked at my coefficients. If I do not apply the summation with constant x. I get the same numbers as you gave (just looked at 4 decimals and handful of coefficients). I think we do the same computation except I changed order of summation. Gottfried RE: A nice series for b^^h , base sqrt(2), by diagonalization  Gottfried  06/11/2009 Fun... I now could make use of the upper (repelling) fixpoint with this type of series. Here we have the upper fixpoint t=4, u=log(4) ~ 1.3862943611... for the same base b = sqrt(2), and we can compute the fractional heights for some appropriate initialvalue x, say x=5: to get context I quote the previous: Gottfried Wrote:Here for base b = sqrt(2) and for brevity v = u^h: and again v for u^h and f(h) for the longish expexpression: with different coefficients. (We have to compute the appropriate Utmatrix and also Wmatrix now) For x=5 we have by fixpointshift x1 = x/t  1 = 5/4  1 = 0.25 and the evaluation of the schröderfunction Code: ´ S~ = V(5/41)~ * W = V(0.25)~ * W gives, by the summation in each column, a set of series, which converge good with 64 terms(n=64 is my selected vector/matrixdimension) and give the vector S containing all results. Then the coefficients from WI (which represent the inverse of schröderfunction) are multiplied into to get the constant vector M, which has now the coefficients which are independent from u^h: Code: ´ M~ = S ~ * diag(WI[,1]) Then the intermediate value y as in the previous msg is again: Thus, having the fixpoint t=4 here, we get the new coefficients for the series from the vector M: Code: ´ f(h) = 4 + 4* y which can be evaluated for some height h, as long as the occuring series (by M~*V(u^h)) converges or can be Eulersummed. This gives the powerseries: For the heights h=0..2 in 1/32steps I get the following values: Code: ´ h  f(h) = exp_sqrt(2)°h(5)= For instance, f(1) should be log(5)/log(b) and we have from the table at h=1 the value 4.64385618977 which agrees with direct computation log(5)/log(sqrt(2)) = 4.64385618977 , and it should be b^f(1.5) = f(0.5), which can be checked easily using values from the table. Don't know yet, whether this has some benefit so far. RE: A nice series for b^^h , base sqrt(2), by diagonalization  Gottfried  06/11/2009 (06/11/2009, 06:26 PM)Gottfried Wrote: Don't know yet, whether this has some benefit so far. It looks, as if we had a discussion of that recently in Upper superexponential I'm excerpting a bit of Henryk's post: (03/29/2009, 11:23 AM)bo198214 Wrote: As it is wellknown we have for the regular superexponential at the lower fixed point. My construction in the previous post was obviously the same as that above construction (*1) ... Gottfried (I added the comments //... ) Gottfried Wrote:Then we can write where the fixpoint "a" is simply given as constant 2 and could be generalized to the symbol. The sumexpression describes the inverse of the schröderfunction chi^1 in Henryk's post. The formula for the repelling fixpoint replaces simply 2 by 4 and (1/21) by (5/41) and uses the adapted schröderfunction. So I think it's useful to redirect replies to the other thread... 