• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 sum of log of eigenvalues of Carleman matrix bo198214 Administrator Posts: 1,595 Threads: 101 Joined: Aug 2007 08/28/2009, 11:27 AM Hey Gottfried, did you notice that the sum over the logarithms of the eigenvalues of the Carleman matrix of exp converge (for increasing matrix size)? Moreover also if you take the n-th power of the logarithms, for any n. This woudl be a direct consequence of the matrix power method (for non-integer iteration of exp) converging to an analytic function. Gottfried Ultimate Fellow Posts: 873 Threads: 128 Joined: Aug 2007 08/28/2009, 08:43 PM (This post was last modified: 08/28/2009, 08:53 PM by Gottfried.) (08/28/2009, 11:27 AM)bo198214 Wrote: Hey Gottfried, did you notice that the sum over the logarithms of the eigenvalues of the Carleman matrix of exp converge (for increasing matrix size)? Moreover also if you take the n-th power of the logarithms, for any n. This woudl be a direct consequence of the matrix power method (for non-integer iteration of exp) converging to an analytic function.Hmm, for dim=8..24 I get them always near null at machine-precision (Pari/GP, 200digit or 800 digits internal prec). That means the product of the eigenvalues is near 1 no matter what dimension I select. There may be an error, however the procedure is simple. Here is the Pari/Gp code Code:fmt(800,12) {for(dim=8,24,    B = VE(fS2F,dim)*VE(P,dim)~ ;  \\ construct the Bell(transp. Carlemann)-matrix for exp(x)    tmpW = mateigen(B);             \\ getting the eigenvectors in tmpW    tmpD=HadDiv(B*tmpW,tmpW)[1,];   \\ getting the eigenvalues in tmpD                                    \\ this is simpler than the "official"                                    \\ method: tmpD = diag(tmpW^-1 * B * tmpW)    sulog = sum(k=1,#tmpD,log(tmpD[k]));    print(dim," ",sulog); )} 8 -3.255463966 E-808 9 -9.22381457 E-808 10 4.88319595 E-808 11 2.821402104 E-807 12 6.51092793 E-808 13 -8.35569084 E-807 14 -1.519216517 E-807 15 3.157800047 E-806 16 -3.393278608 E-805 17 -5.444003874 E-804 18 -4.808428794 E-804 19 2.106242431 E-801 20 -1.177930451 E-800 21 -2.356690583 E-799 22 -3.10781078241 E-798 23 -3.76652961891 E-797 24 -3.06512885635 E-797 Example eigenvalues for dim=24 [4.28673736924 E-11] [0.00000000296568145370] [0.0000000957784063100] [0.00000191766745327] [0.0000266643844037] [0.000273351215354] [0.00214050544928] [0.0130807983589] [0.0630992348914] [0.240819894743] [0.729062578542] [1.00000000000] [1.89149672765] [5.08315258442] [15.5889319581] [54.8943446446] [221.893591029] [1035.09334661] [5636.83816890] [36538.7788311] [290981.552989] [3004492.63267] [44636646.3387] [1247092190.35] Wouldn't say, this is exactly convergence with increasing dimension... ;-) Gottfried Gottfried Helms, Kassel bo198214 Administrator Posts: 1,595 Threads: 101 Joined: Aug 2007 08/28/2009, 08:58 PM Hm, I dont know whether it has a deeper meaning its just an observation Gottfried Ultimate Fellow Posts: 873 Threads: 128 Joined: Aug 2007 08/28/2009, 09:10 PM (08/28/2009, 08:58 PM)bo198214 Wrote: Hm, I dont know whether it has a deeper meaning its just an observation Hmm, this means, the determinant of this matrix is 1 for finite dimension - which extends then also for iterates/powers. That the determinant is 1 for finite dimension results also from the determinants of its factors, the stirling- ind the bpascal-matrix. Both are triangular and have units on the diagonal, so det(fS2F) = det(P) = 1 and det(B) = det(fS2F * P~) = det(fS2F)*det(P~) = 1 *1. Gottfried Helms, Kassel Gottfried Ultimate Fellow Posts: 873 Threads: 128 Joined: Aug 2007 08/28/2009, 09:34 PM (08/28/2009, 09:10 PM)Gottfried Wrote: (08/28/2009, 08:58 PM)bo198214 Wrote: Hm, I dont know whether it has a deeper meaning its just an observation Hmm, this means, the determinant of this matrix is 1 for finite dimension - which extends then also for iterates/powers. That the determinant is 1 for finite dimension results also from the determinants of its factors, the stirling- ind the bpascal-matrix. Both are triangular and have units on the diagonal, so det(fS2F) = det(P) = 1 and det(B) = det(fS2F * P~) = det(fS2F)*det(P~) = 1 *1.It is also interesting in contrast to the version, which has the powerseries developed at the first complex fixpoint for exp(x), x0 = 0.318131505205 + 1.33723570143*I . With dim=64 I get -at least for the integer iterates -1,1,2 the expected values to 10 digits accuracy - however, the sum of the logarithms of the eigenvalues should be simply, but consequently, log(x0^0) + log(x0^1) + log(x0^2) + ... = log(x0)*(0 + 1 + 2 + 3 + ...) =???= log(x0) * zeta(-1) =-0.0265109587671 - 0.111436308453*I .... well, better not to ride the horse to death... Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post A support for Andy's (P.Walker's) slog-matrix-method Gottfried 4 6,018 03/08/2021, 07:13 PM Last Post: JmsNxn New Quantum Algorithms (Carleman linearization) Finally Crack Nonlinear Equations Daniel 2 2,061 01/10/2021, 12:33 AM Last Post: marraco Tommy's matrix method for superlogarithm. tommy1729 0 3,793 05/07/2016, 12:28 PM Last Post: tommy1729 Regular iteration using matrix-Jordan-form Gottfried 7 15,837 09/29/2014, 11:39 PM Last Post: Gottfried Q: Exponentiation of a carleman-matrix Gottfried 0 4,565 11/19/2012, 10:18 AM Last Post: Gottfried "Natural boundary", regular tetration, and Abel matrix mike3 9 25,186 06/24/2010, 07:19 AM Last Post: Gottfried spectrum of Carleman matrix bo198214 3 8,165 02/23/2009, 03:52 AM Last Post: Gottfried Matrix Operator Method Gottfried 38 77,669 09/26/2008, 09:56 AM Last Post: Gottfried matrix function like iteration without power series expansion bo198214 15 36,009 07/14/2008, 09:55 PM Last Post: bo198214 Eigenvalues of the Carleman matrix of b^x Gottfried 7 14,119 06/29/2008, 12:06 PM Last Post: bo198214

Users browsing this thread: 1 Guest(s)