Re: Tetration again!

Am 23.12.2007 20:57 schrieb mike3:
On Dec 23, 5:22 am, Gottfried Helms <he...@xxxxxxxxxxxxx> wrote:
Am 23.12.2007 09:48 schrieb mike3:

Hmm. However I still don't seem to get how to compute
the polynomials. Is there an "easy" (simple) algorithm for
the coefficients of the powers of h in those polynomials?
Hmm, if you're able to find an easier, for instance recurrence,
formula - this would be a big shot, I'd guess...


Well, maybe then there wasn't an "easy" algorithm, but how does
one do the "hard" algorithm? How do you use that matrix to get
the polynomials? I didn't seem to see anything in the link you
posted that showed explicitly how to derive the polynomials from
that matrix.
a) set (for instance length 6) series = [1,b,c,d,e,f]
So you have
f(x) = 1x + bx^2 + cx^3 + dx^4 +ex^5 + fx^6
Next, f(f(x)) is
f(f(x)) = 1 f(x) + b f(x)^2 + c f(x)^3 + ...
expand, and collect terms according to like powers of x
Iterate until you have enough iterations.
It's surely too much work for paper&pen if more than 3 or
4 elements are involved, but Pari/Gp does it for you for more
elements.
If b,c,d,e,f are numeric (b=1/2!, c=1/3!, d=1/4!,...), then
Pari did it for me up to 64 terms (using 32 Mb memory).
Then the polynomials are computed by polinterpolate()
on the sequence of terms. For the first rows the computation
of the polynomials is easy with the help of a difference
table.

b) define S2 up to a certain dimension, say 32 or 64.
Then S2 = matrix(32,32,r,c, [r,c]/r!), where [r,c] are the
Stirling-numbers 2'nd kind (version of wikipedia).
Any matrix-able program may then compute the powers of S2,
and in the 2'nd columns of the powers of S2 there are the
same coefficients as mentioned above, which can then be
used for polinterpolate().

I don't have a more compact routine, yet...

Gottfried

--
---

Gottfried Helms, Kassel
.

Relevant Pages

  • Re: An observation of Weils
    ... Do you know how to use the Euclidean algorithm to compute the ... polynomials in one variable, because we have a division algorithm ... there and keep all the equations and coefficients integral. ... Of course if pand qhave no common factors in the first ...
    (sci.math)
  • Re: Finding the Formula...
    ... algorithm for generating the coefficients. ... how you said it took many many iterations with huge matrices to ... If my "hunch" is right and the polynomials are of degree n, ...
    (sci.math)
  • Re: An observation of Weils
    ... homogeneous polynomials with integer coefficients, ... You need the division algorithm, that is, you need to know ... But any common divisor will, ...
    (sci.math)
  • Re: An observation of Weils
    ... homogeneous polynomials with integer coefficients, ... the greatest common divisor d of two integers p,q ... You need the division algorithm, that is, you need to ...
    (sci.math)
  • Re: sparse polynomial arithmetic
    ... If FLINT neither uses machine integer/float arithmetic, ... It doesn't multiply individual coefficients at all for polynomials ... If your algorithm is going to be fairly general, ...
    (sci.math.symbolic)