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short compilation:fractional iteration-> eigendecomposition
#7
bo198214 Wrote:What I miss however is a suitable discussion of finite versus infinite matrices. For example if you approximate an infinite matrix M by finite matrices M_n then the inverse of the infinite matrix is not always the limit of the inverses of M_n.

Henryk -

I've one example loosely related to this: non-uniqueness of reciprocal.

Let base t=e, then the formal powerseries for f_t(x)=log(1+x)/log(t) is that of f(x) = log(1+x) and has the coefficients C_0=[0,1,-1/2,+1/3,-1/4,...]
Using them to construct the matrix-operator S1, we get the well known, infinite sized triangular matrix of Stirling-numbers 1'st kind (with factorial similarity scaling) S1; whose reciprocal is that of Stirling-numbers 2'nd kind, analoguously scaled.

I tried to include the property of multivaluedness of general logarithms, which is log(1+x) = y + k*2*Pi*i =y + w_k by replacing the leading zero in the above set of coefficients to obtain C_k=[w_k,1,-1/2,+1/3,-1/4,...]
I generated the according matrix-operator S1_k based on this formal powerseries.

Although we discuss theoretically infinite matrices the finite truncation of this made sense for k=1 and 2, so my approximations for S1_1 and S1_2 "worked" as expected (using sizes up to 64x64) :

I got, with good approximation to about 12 visible digits, the expected complex-valued logarithms, and even the reciprocity/inverse-conditions S1_1*S2 = I and even S1_2*S2 = I held (this was surely expected but was still somehow surprising Smile )

Anyway - I'd like to see more examples for problems with the infinite-size-inverse of triangular matrix-operators to get an idea about the basic characteristics of those problems. Do you know some?

Gottfried
Gottfried Helms, Kassel
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RE: short compilation:fractional iteration-> eigendecomposition - by Gottfried - 01/20/2008, 02:32 PM

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