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 Tetra-series Gottfried Ultimate Fellow Posts: 766 Threads: 119 Joined: Aug 2007 10/31/2009, 10:33 PM (This post was last modified: 10/31/2009, 10:39 PM by Gottfried.) (10/31/2009, 09:37 PM)andydude Wrote: $F = \mathbf{B}^{-1}G$ and when the matrix size is even I get the first series, and when the matrix size is odd, I get the second series. Hmm, I didn't catch the actual computation, but that's possibly not yet required. I guess, you need the technique of divergent-summation ; it may be, that the implicite series, which occur by multiplication of $\mathbf{B}^{-1}G$, have alternating signs, are not converging well, or even are divergent. So you could include the process of Euler-summation. I found a very nice method to get at least an overview, whether such matrix-product suffers from non-convergence (but which is reparable by Cesaro/Euler-summation). I've defined a diagonal-matrix dE(o) of coefficients for Euler-summation of order o, which can simply be included in the matrix-product. Write $\mathbf{B}^{-1} * dE(o) * G$ where o=1.0 o=1.5 or o=2. With too small o the implicite sums in matrix-multiplication begin to oscillate from some terms (order is "too weak") , for too high o the oscillation is so heavily suppressed, that with dim-number of terms the series is not yet converging. In a matrix-product using dimension dim the number of dim^2 of such sums occur. While likely not all that sums can be handled correctly by that same Euler-vector, for some of them you will see a well approximated result and a general smoothing, making the result matrix-size independent, especially averages between size dim and size dim+1. In my implementation order o=1 means no change; simply dE(1) = I ; dE(1.7)..dE(2.0) can sum 1-1+1-1... and similar, dE(2.5)..dE(3.0) can sum divergence of type 1-2+4-8+...-... and so on (details for toy-implementation in Pari/GP see below, I can send you some example scripts if this is more convenient) Gottfried Code://  Pari/GP \\ a vector of length dim returning coefficients for Euler-summation of \\ order "order" (E(1) gives the unit-vector:direct summation {E(order, dim=n) = local(Eu); Eu=vector(dim);Eu[1]=order^(dim-1); for(k=2,dim,Eu[k]=Eu[k-1]-(order-1)^(dim-k+1)*binomial(dim-1,k-2)); Eu=Eu/order^(dim-1); return(Eu);} \\ returns this as diagonal-matrix dE(order,dim=n) = return( matdiagonal(E(order,dim)) ) Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Tetra-series - by Gottfried - 11/20/2007, 12:47 PM RE: Tetra-series - by andydude - 11/21/2007, 07:14 AM RE: Tetra-series - by Gottfried - 11/22/2007, 07:04 AM RE: Tetra-series - by andydude - 11/21/2007, 07:51 AM RE: Tetra-series - by Gottfried - 11/21/2007, 09:41 AM RE: Tetra-series - by Ivars - 11/21/2007, 03:58 PM RE: Tetra-series - by Gottfried - 11/21/2007, 04:37 PM RE: Tetra-series - by Gottfried - 11/21/2007, 06:59 PM RE: Tetra-series - by andydude - 11/21/2007, 07:24 PM RE: Tetra-series - by Gottfried - 11/21/2007, 07:49 PM RE: Tetra-series - by andydude - 11/21/2007, 08:39 PM RE: Tetra-series - by Gottfried - 11/23/2007, 10:47 AM RE: Tetra-series - by Gottfried - 12/26/2007, 07:39 PM RE: Tetra-series - by Gottfried - 02/18/2008, 07:19 PM RE: Tetra-series - by Gottfried - 06/13/2008, 07:15 AM RE: Tetra-series - by Gottfried - 06/22/2008, 05:25 PM Tetra-series / Inverse - by Gottfried - 06/29/2008, 09:41 PM RE: Tetra-series / Inverse - by Gottfried - 06/30/2008, 12:11 PM RE: Tetra-series / Inverse - by Gottfried - 07/02/2008, 11:01 AM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 10:38 AM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 11:01 AM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 01:25 PM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 02:40 PM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 09:37 PM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 10:33 PM RE: Tetra-series / Inverse - by Gottfried - 11/01/2009, 07:45 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 03:56 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 04:12 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 05:04 AM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 12:58 PM

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