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 [Update] Comparision of 5 methods of interpolation to continuous tetration Gottfried Ultimate Fellow Posts: 854 Threads: 126 Joined: Aug 2007 10/15/2013, 11:14 PM (This post was last modified: 10/15/2013, 11:17 PM by Gottfried.) Hi Sheldon, first to your second question, number 1): yes, the function is real on the real axis because we use just the eigendecomposition of a real matrix - no complex coefficients anywhere. About number 2) I've no idea yet, perhaps its too late for me tonight... I'll come back to this tomorrow. We do not have a true Schröder- and inverse Schröder-function because the diagonal after the diagonalization (the set of eigenvalues) does not contain the consecutive powers of the log of the fixpoint but are more or less arbitrary - depending on the size of the matrix. So I don't see at the moment how to model the sexp() and inverse sexp()-functionality. Gottfried Here is code to reproduce the behaviour - no optimization is made, just the basic functionality. If you want to exceed the range of convergence of the "power series" you should add explicite iteration over the integer-part of the heights. Code:n=24   \\ set size for matrices, note this needs ~ 200 digits real precision,        \\ n=64 needs about 4000 digits precision and 2 hours to be computed! default(precision,400) \\ when n~ 24 \\ procedures =========================================================== \\    several global variables/matrices will be created/changed {init(base,matsize,prec) = local(a);    n=matsize;  \\ set size for matrices, note size=24x24 needs ~ 200 digits real precision,                \\ n=64 needs about 4000 digits precision and 2 hours to be computed!    default(precision,prec,1); \\ when n~ 24 then prec ~ 200       b = base;bl=log(b); \\ sets the base for tetration x-> 4^x                     \\ bl must have the required precision depending on matrix-size n                     \\ thus must be recomputed if size and precision is increased!       B = matrix(n,n,r,c,((c-1)*bl)^(r-1)/(r-1)!) ;                  \\ the Carlemanmatrix for x-> b^x                  \\ the resulting coefficients for the power-series                  \\ are in the second column      \\ Diagonalization such that B = M * D * W (where W=M^-1)       M = mateigen(B);       W = M^-1 ;       D = diag(W * B * M);       POLY= M * matdiagonal(W[,2])  \\ this matrix is a "kernel" which needs only the parameter                     \\ for the height to give the associated power series       print("matrices are created");    return(1);} f(x) = sum(k=0,n-1,x^k * B[1+k,2])  \\ just to show how this works for the                                     \\ function f(x) itself         \\ then we can the h'th fractional power  of Bb approximate by M * D^h * W (= B^h)          make_powerseries(h) = return( POLY * vectorv(n,r, D[r]^h) ); { tet_polynomial(x,h) = local(pc);        pc = POLY*vectorv(n, r , D[r]^h ) ;  \\ creates coefficients for power series        result = sum(k=0,n-1, x^k * pc[1+k]);  \\ evaluates the powerseries        return(result);} \\ note that M and W do not really implement the Schröder-functions because \\ the entries in D are not consecutive powers of one argument \\ then you start:================================================================= init(4,24,200) \\ for matrixsize 24x24 you'll need at least 200 digits precision x0 = 0.1*I x05= tet_polynomial( x0 , 0.5 )   \\ halfiterate from x0=0.1*I x1 = tet_polynomial( x05, 0.5 )   \\ halfiterate from x05 should equal one whole iteration: x1 - b^x0    \\ check the difference   \\ bigger matrixsize init(4,32,600) \\ for matrixsize 32x32 you'll need at least 600 digits precision x0 = 0.1*I x05= tet_polynomial( x0 , 0.5 )   \\ halfiterate from x0=0.1*I x1 = tet_polynomial( x05, 0.5 )   \\ halfiterate from x05 should equal one whole iteration: x1 - b^x0    \\ check the difference   \\ bigger matrixsize init(4,64,4000) \\ for matrixsize 64x64 you'll need at least 4000 digits precision                  \\ this needs 2 hours for computation! x0 = 0.1*I x05= tet_polynomial( x0 , 0.5 )   \\ halfiterate from x0=0.1*I x1 = tet_polynomial( x05, 0.5 )   \\ halfiterate from x05 should equal one whole iteration: x1 - b^x0    \\ check the difference Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/13/2013, 02:20 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by MikeSmith - 10/14/2013, 10:42 AM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/14/2013, 01:19 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by MikeSmith - 10/14/2013, 08:22 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/15/2013, 12:59 AM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/14/2013, 10:07 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/15/2013, 12:00 AM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/15/2013, 01:09 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/15/2013, 07:14 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/15/2013, 11:14 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/16/2013, 12:54 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/16/2013, 04:12 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/16/2013, 05:07 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 10/22/2013, 12:17 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/22/2013, 01:53 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/27/2013, 11:40 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/28/2013, 04:17 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 10/28/2013, 11:11 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 10/28/2013, 11:29 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/29/2013, 09:37 AM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 10/28/2013, 11:32 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 10/29/2013, 01:11 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by sheldonison - 10/29/2013, 02:52 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 11/01/2013, 02:16 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 02/01/2014, 12:09 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 02/04/2014, 12:31 AM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 02/03/2014, 01:13 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 02/03/2014, 01:44 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 02/03/2014, 01:59 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by tommy1729 - 02/03/2014, 10:35 PM RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 02/03/2014, 11:07 PM

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