Matrix-method: compare use of different fixpoints

[Gottfried]

Seems we have a basic misunderstanding here, though I dont know where yet. Ok, let me nail the facts:

We have something that is called the matrix operator method, this truncates \( B_b \) (the Carleman matrix of \( b^x \)) to n
then decomposes uniquely via Eigenvalues \( {B_b}_{|n}=W_{|n} D_{|n} W_{|n}^{-1} \). And then defines \( {B_b}^t = \lim_{n\to\infty} W_{|n} {D_{|n}}^t W_{|n}^{-1} \). And we get the coeffecients of \( {\exp_b}^{\circ t} \) from the first column of \( B_b \).
I want your acknowledgement on this.

Well, you say it in the next sentence: let truncation aside. In all my considerations truncation always meant only the need for practical implementation and thus assumes approximative results (however good and extrapolatable for the infinite-thought series). If I want to employ the exponential-series to compute e^x I need the notion of infinitely many terms and in turn, if expressed as a vector/vector or vector/matrix-multiplication of inherently infinite size. I never thought, that a/the matrix-method could reduce the problem to one which requires only finitely truncated series.

But perhaps you have to explain in more detail what you mean by

based on an analytical description of each entry, then I actually work with finite truncations of an assumed infinite matrix, which may provide multiple solutions for the same composed theoretical result matrix.

I observed a certain structure, which the eigenvalues of the most stable matrices Bs showed. From there I took my first hypothese about the structure of the set of eigenvalues: they form the powerseries of the u =log(t) of the first fixpoint t (base b = t^(1/t)) (where the notion of "fixpoint" was unfamiliar to me and never used it in the beginning).

Second I observed other structures in the empirical eigen-matrices, which stabilized with higher dimensions and suggested some basic assumptions about the composition of some entries, for instance the first two rows. So I had a hypothese about an analytical description of the form of each entry wi(r,c) in WI = W^-1 of the first two rows, say

wi(0,c) = t^c
wi(1,c) = binomial(c,1)*t^c

I could then verify this as a possibility by setting

WI[,0] * Bs = u^0 * WI[,0]
and
WI[,1] * Bs = u^1 * WI[,1]
by simple elementary consideration of the implicte exponential series.

Then the next empirical rows in WI seemed to approximate (and stabilize with higher dimension) to some composition of binomially weighted powerseries coefficients. By considering some possibilities for the entries in wi(2,c) it seemed they were binomially weighted compositions like wi(0,c) and wi(1,c), though a bit more complicated.

In short, with some more hypotheses and analytical considerations I got finally an idea, how I could find such compositions for all subsequent rows. This needed then an eigensystem-related computation, but only of a triangular system, which can be solved using finitely many terms for each wi(row,col) . It occured, that WI was composed by a matrix-multiplication of some XI by P~ , where P is a constant(the pascal-matrix) and XI is triangular.

Finally I had a process, which can compute the composition of each entry xi(r,c) in XI as a finite two-variable polynomial of the formal parameters u and t, where the order of t is the same as the row-index and the order of u is binomial(rowindex,2) (or opposite, don't have it at hand). Thus I have a description of XI, independent of the size of the matrix/its truncation parameter, which together with P~ give a matrix W^-1, which satisfies the properties of an eigenmatrix for Bs together with a diagonal D containing the consecutive powers of u - and: for the assumption of infinite size.

Since XI=X^-1 is triangular, also each entry in X can be computed exactly from this, and I have implicitely the formal description of the polynomials in their parameters (t,u) of the triangular core X*D*X^-1 , and could store this polynomial descriptions for each term in a database... I'm sometimes working on this a bit, but the polynomials (and thus the memory for its representation) of the individual terms grow quadratically with its row-index, so actually I still do the whole computation of terms on the fly with the current numerical parameters numerically.

That means: if I speak about the eigensystem W,D,W^-1 I do not speak about the eigensystem of a truncated matrix (as retrieved by a numerical eigensystem-solver) but about the entries of an infinite eigenmatrix whose infinite extension expresses the required powerseries for the tetration-operation (and of which the truncation is then comparable with the truncation that we use, if we compute, say, exp(x)) and their formal description in terms of the parameters t and u.

And we have to think about a name for this additional method.

Ok, if it introduces non-uniqueness now, and it seems to be a rather private, say "my matrix method", what do you think of "my-oh-my" (or mei-o-mei)-method ? ;-)

Gottfried