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Taylor polynomial. System of equations for the coefficients.
#11
(05/05/2015, 07:40 AM)Gottfried Wrote: P*A = A*Bb

I think that we are speaking of different things.

Obviously, there should be a way to demonstrate the equivalence of both, because they are trying to solve the same problem; looking for the same solution.

But as I understand, the Carleman matrix A only contains powers of a_i coefficients, yet if you look at the red side, it cannot be written as a matrix product A*Bb, because it needs to have products of a_i coefficients (like ). Maybe it is a power of A.Bb, or something like A^Bb?

The Pascal matrix on the blue side is the exponential of a much simpler matrix



Maybe the equation can be greatly simplified by taking a logarithm of both sides.
I have the result, but I do not yet know how to get it.
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#12
(05/06/2015, 02:42 PM)marraco Wrote:
(05/05/2015, 07:40 AM)Gottfried Wrote: P*A = A*Bb

I think that we are speaking of different things.

Obviously, there should be a way to demonstrate the equivalence of both, because they are trying to solve the same problem; looking for the same solution.

But as I understand, the Carleman matrix A only contains powers of a_i coefficients, yet if you look at the red side, it cannot be written as a matrix product A*Bb, because it needs to have products of a_i coefficients (like ). Maybe it is a power of A.Bb, or something like A^Bb?

No, no ... In your convolution-formula you have in the inner of the double sum powers of powerseries (the red-colored formula in your first posting ) with the coefficients of the a()-function (not of its single coefficients), and if I decode this correctly, then this matches perfectly the composition of

V(x)*A * Bb = (V(x)*A) * Bb = [1,a(x), a(x)^2, a(x)^3),...] * Bb = V(a(x))*Bb

Only, that after removing of the left V(x)-vector we do things in different order:

V(x)*A * Bb = V(x)*(A * Bb )

and I discuss that remaining matrix in the parenthese of the rhs. That V(x) can be removed on the rhs and on the lhs of the matrix-equation must be justified; if anywhere occur divergent series, this becomes difficult, but as far as we have nonzero intervals of convergence for all dot-products, this exploitation of associativity can be done /should be possible to be done (as far as I think). (The goal of this all is of course to improve computability of A, for instance by diagonalization of P or Bb and algebraic manipulations of the occuring matrix-factors).

Anyway - I hope I didn't actually misread you (which is always possible given the lot of coefficients... )

Gottfried
Gottfried Helms, Kassel
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#13
I misinterpreted what the Carleman matrix was. I tough that it contained the powers of the derivatives of a function (valued at zero), but it contains the derivatives of the powers of a function, so it actually haves the products of the aᵢ coefficients (of bᵢ in your notation).

________________

I tried to use this method to find the coefficients for exponentiation: bˣ=Σbᵢ.xⁿ

The condition is
b.(x+1)=b.Σbᵢ.xⁿ

which translates into
P.[bᵢ]=b.[bᵢ]

or
[P-b.I].[bᵢ]=0

The solution should be bᵢ=ln(b)ⁱ / i!

I found bᵢ=c. (ln(b)ⁱ/i!), where c is an arbitrary constant, because, obviously c.b⁽ˣ⁺¹⁾=b.(c.bˣ)

I was bugged for the fact that any equation for solving tetration I tried seems to have at least one degree of liberty. I think now that it should be explained by one (at least) arbitrary constant in the solution.

This looks analogous to constants found in the solution of differential equations, so I wonder if the evolvent of the curves generated by the constant is also a solution, and what is his meaning.
I have the result, but I do not yet know how to get it.
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#14
So, we want the vector , from the matrix equation:



where "r" is the row index of the first matrix at left, and "i" his column index.
Note that in the last equation, both r and i start counting from zero for the first row and column.


______________________________________
P(i) is the partition function

The first few values of the partition function are (starting with p(0)=1):

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, … (sequence A000041 in OEIS; the link has valuable information about the partition function).


______________________________________
is the number of repetitions of the integer j in the partition of the number i



______________________________________
Solving the equation

If we do the substitution , we simplify the first equation to:




______________________________________
Special base.

This equation suggest a special number, which is m=1.7632228343518967102252017769517070804...
m is defined by

For the base a=m, the equation gets simplified to:



But let's forget about m for now.

______________________________________

We are now very close to the solution. The only obstacle remaining is the product:


If we can do a substitution that get us rid of him, we have the solution:



At this point we only need to substitute , where f is arbitrary, to get:





... and we get:



The choice of f, very probably, determines the value for °a, and the branch of tetration.

(01/03/2016, 11:24 PM)marraco Wrote: [Image: lynxfBI.jpg?1]
I have the result, but I do not yet know how to get it.
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#15
^^ Sorry. I made a big mistake. We cannot substitute of course.

Maybe would work as an approximation, because we know that tends very rapidly to a line on logarithmic scale. Anyways, it would be of little use.

We know that the are the derivatives of , so a Fourier or Laplace transform would turn the derivatives into products. But that would mess with the rest of the equation.
I have the result, but I do not yet know how to get it.
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#16
Here I make an expansion of a row, in hope that it helps somebody to digest the equation.

(01/13/2016, 04:32 AM)marraco Wrote: We are now very close to the solution. The only obstacle remaining is the product:

The product is what I called "the integer divisor"

(05/03/2015, 04:35 AM)marraco Wrote:




^^ Here I expanded the row for i=9 of the equation:




after the substitution :


The problematic terms come from the factors . The q! divisors may emerge not from the term raised to q. q! could emerge from the absence of the other terms: .
For example, the term is actually

I have the result, but I do not yet know how to get it.
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#17
(01/13/2016, 04:32 AM)marraco Wrote: So, we want the vector , from the matrix equation:


Thanks to Daniel advice, is easy to see that the red side can be derived as a direct application of Faà di Bruno's formula



in the blue red equation,

(04/30/2015, 03:24 AM)marraco Wrote: We want the coefficients aᵢ of this Taylor expansion:



They should match this equation:
is easy to see on the red side, that









I have the result, but I do not yet know how to get it.
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#18
(05/01/2015, 01:57 AM)marraco Wrote: This is a numerical example for base a=e
(...)

So you get this systems of equations (blue to the left, and red to the right):


Code:
[1 1 1 1 1  1  1  1  1]    [ 1]   [e]
[0 1 2 3 4  5  6  7  8]    [a₁]   [e.a₁]
[0 0 1 3 6 10 15 21 28]    [a₂]   [e.a₂+e/2.a₁²]
[0 0 0 1 4 10 20 35 56]    [a₃]   [e.a₃+e.a₁.a₂+e/6.a₁³]
[0 0 0 0 1  5 15 35 70]  * [a₄] = [e.a₄+e/2.a₂²+e.a₃.a₁+e/2.a₂.a1²+e/24.a₁⁴]
[0 0 0 0 0  1  6 21 56]    [a₅]   [...]
[0 0 0 0 0  0  1  7 28]    [a₆]   [...]
[0 0 0 0 0  0  0  1  8]    [a₇]   [...]
[0 0 0 0 0  0  0  0  1]    [a₈]   [...]


Quote:It is a non linear system of equations, and the solution for this particular case is:
Code:
a₀=    1,00000000000000000
a₁=    1,09975111049169000
a₂=    0,24752638354178700
a₃=    0,15046151104294100
a₄=    0,12170896032120000
a₅=    0,16084324512292400
a₆=    -0,02254254634348470
a₇=    -0,10318144159688800
a₈=    0,06371479195361670
(...)
This is perhaps a good starting point to explain the use of Carleman-matrices in my (Pari/GP-supported) matrix-toolbox, because you've just applied things analoguously to how I do this, only you didn' express it in matrix-formulae.
To explain the basic idea of a Carlemanmatrix:
consider a powerseries


We express this in terms of the dot-product of two infinite-sized vectors

where the column-vector A_1 contains the coefficients and the row-vector
Now to make that idea valuable for function-composition /- iteration it would be good, if the output of such an operation were not simple a scalar, but of the same type ("vandermonde vector") as the input ( ).

This leads to the idea of Carlemanmatrices: we just generate the vectors where the vector contains the coefficients for powers of f(x), such that
... in a matrix getting the operation:
or


Having this general idea we can fill our toolbox with Carlemanmatrices for the composition of functions for a fairly wide range of algebra.
For instance the operation INC

and its h'th iteration ADD

is then only a problem of powers of P

The operation MUL needs a diagonal vandermonde vector:


The operation DEXP (= exp(x)-1) needs the matrix of Stirlingnumbers 2nd kind, similarity-scaled by factorials:


and as an exercise, we see, that if we right-compose this with the INC -operation, we get the ordinary EXP operator, for which I give the matrix-name B:



Of course, iterations of the EXP require then only powers of the matrix B.

To see, that this is really useful, we need a lemma on the uniqueness of power-series. That is, in the new matrix-notation:
If a function is continuous for a (even small) continuous range of the argument x, then the coefficients in A_1 are uniquely determined.
That uniqueness of the coefficients in A_1 is the key, that we can look at the compositions of Carleman-matrices alone without respect of the notation with the dotproduct by V(x) and for instance, we can make use of the analysis of Carlemanmatrix-decompositions like

and can analyze

directly, for instance to arrive at the operation LOGP : log(1+x)


<hr>
Now I relate this to that derivation which I've quoted from marraco's post.


(05/01/2015, 01:57 AM)marraco Wrote: This is a numerical example for base a=e
(...)
So you get this systems of equations (blue to the left, and red to the right):
Code:
[1 1 1 1 1  1  1  1  1]    [ 1]   [e]
[0 1 2 3 4  5  6  7  8]    [a₁]   [e.a₁]
[0 0 1 3 6 10 15 21 28]    [a₂]   [e.a₂+e/2.a₁²]
[0 0 0 1 4 10 20 35 56]    [a₃]   [e.a₃+e.a₁.a₂+e/6.a₁³]
[0 0 0 0 1  5 15 35 70]  * [a₄] = [e.a₄+e/2.a₂²+e.a₃.a₁+e/2.a₂.a1²+e/24.a₁⁴]
[0 0 0 0 0  1  6 21 56]    [a₅]   [...]
[0 0 0 0 0  0  1  7 28]    [a₆]   [...]
[0 0 0 0 0  0  0  1  8]    [a₇]   [...]
[0 0 0 0 0  0  0  0  1]    [a₈]   [...]


First we see the Pascalmatrix P on the lhs in action, then the coefficients of the Abel-function in the vector, say, A_1 . So the left hand is


To make things smoother first, we assume A as complete Carlemanmatrix, expanded from A_1. If we "complete" that left hand side to discuss this in power series we have


It is very likely, that the author wanted to derive the solution for the equation ; so we would have for the right hand side

and indeed, expanding the terms using the matrixes as created in Pari/GP with, let's say size of 32x32 or 64x64 we get very nice approximations to that descriptions in the rhs of the quoted matrix-formula.

What we can now do, depends on the above uniqueness-lemma: we can discard the V(x)-reference, just writing and looking at the second column of B only we get as shown in the quoted post.

So indeed, that system of equations of the initial post is expressible by

and the OP searches a solution for A.

<hr>
While I've -at the moment- not yet a solution for A this way, we can, for instance, note that if A is invertible, then the equation can be made in a Jordan-form:

which means, that B can be decomposed by similarity transformations into a triangular Jordan block, namely the Pascalmatrix - and having a Jordan-solver for finite matrix-sizes, one could try, whether increasing the matrix-sizes the Jordan-solutions converge to some limit-matrix A.

For the alternative: looking at the "regular tetration" and the Schröder-function (including recentering the powerseries around the fixpoint) one gets a simple solution just by the diagonalization-formulae for triangular Carlemanmatrices which follow the same formal analysis using the "matrix-toolbox" which can, for finite size and numerical approximations, nicely be constructed using the matrix-features of Pari/GP.



Gottfried Helms, Kassel
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