(08/05/2014, 12:35 PM)nuninho1980 Wrote::-) Well, I said the technique likely approximates Kneser if the matrix size goes to infinity. However, 32x32 requires 2500 dec digits internal precision to compute the eigen-decomposition - to arrive at about 6 to 8 digits accuracy for the result... Thus I said, this is a fairly rough method.(08/04/2014, 11:10 PM)Gottfried Wrote: With a very rough approximation I get 44.05666 (using Eigendecomposition of truncated matrix 32x32, Pari/GP, 2500 digits internal precision required for this).I'm interested. You can get ~44... Using Kneser is serious!?

If I repeat that I arrive at something like (10^10)*1.004, so the value should be slightly too high, because of course I should arrive at (10^10).

(Note that this method possibly approximates the Kneser solution when the matrix-size is increased without bound)

Gottfried

The procedere is simple: let b be the basis so b=10^10, the log of b bl=10*log(10).

Then define the coefficients for the exponential-series with basis b, which is bseries= sum(k=0,31, bl^k / k! *x^k ) + O(x^32)

Then having the (truncated) exponential-series you can use its coefficients and that of the powers of bseries to form the Carleman-matrix C (see Wikipedia for this or my early postings here, where I did not know the name "Carleman-matrix" but simply called them "Matrix-operator")

Then C provides the coefficients for the formal power-series for b^x, the second power C^2 the coefficients for the formal power series for b^b^x and so on. See the columns of the (top-left segment of) matrix C:

In the first column we have the coefficients for (b^x)^0; in the second that for (b^x)^1 , in the third that for (b^x)^2 and so on. To check the coefficients in the columns just type (10^10)^x and ((10^10)^x)^2 and ((10^10)^x)^3) into Pari/GP.

But while the integer powers of C (and thus the integer iterates of b^x) can easily be computed, fractional powers need eigen-decomposition, such that C = M*D*W (where W=M^-1) and M, D and W must be found by some Eigensystem-solver (for instance using Pari/GP) The crazy observation is, that you need everything with 2500 digits precision if you only use matrixsize 32x32 (matrix-size 16x16 can be made with 200 digits internal precision) for the roots of the characteristic polynomial and the according eigenvectors.

But luckily, because D is diagonal by construction of the eigen-solver, you can apply fractional powers to it by applying fractional powers to its (scalar) diagonal elements. Then, for the half-iterate, you do for instance

C05 = M * D^0.5 * W

and C05 is the Carlemanmatrix for the half iterate of b^x. (Usually we use actually x=1 if we write b^^0.5 for instance)

The top-left segment of C05 based on 32x32-truncation looks like:

and in the second column you find the coefficients for the (approximate) power series (in fact, it is now a polynomial of order 32) for the half iterate of b^x, let's call the function g(x). The occurence suggests, that the coefficients diminuish with higher index, but this might be an artifact because of the truncation, and if I could compute 64x64 it might be that the coefficients still grow higher. But also it might be that the approximation/the overall pattern is already well; and this is backed by the observation, that g(g(x)) approx b^x for small enough x.

If you're really interested I could send you a small collection of Pari/GP-routines to work with this method. Also, for some impression for the compatibility of the diagonalization of the truncated matrixes with the Kneser-method see my small compilation

http://go.helms-net.de/math/tetdocs/Comp...ations.pdf

Have fun -

Gottfried

Gottfried Helms, Kassel