01/24/2011, 08:56 AM
Mike -
without having it definite, I think that your recursive description should be the same as that in the eingenvalue/eigensystem-computation in the APT-article: it is just a simple procedure and works fast.
I was as well looking for a simpler, non-recursive description But I didn't find anything convincing. The only thing I found was that method which solves linear equation systems using the (u-1)(u^2-1)...() - denominators and assumes that at integer heights the numerator must be evenly divisible by that denominator plus the assumption, that the first and last row of the coefficients-matrices are the Stirlingnumbers first and second kind, respectively (or opposite). With this each coefficient-matrix can be determined separately, nonrecursive at least in the sense that the coefficients-matrices (and thus the computation of their second and their last columns) are independent of each other. (Don't know whether the description of this in the mentioned APT-article was too short/imprecise?) I didn't proceed there - it seemed to be too complicated and does not give an optimization over the efficient recursive procedure ; maybe you could find an improvement once the system of conditions is formally stated in terms of linear equations/matrix-formula and thus possibly get a more explicite expression in terms of the "curlybraces" ;-) only.
Well, I'll give it another try myself today, but my emphasis on this has a bit cooled down, to be frank...
Gottfried
without having it definite, I think that your recursive description should be the same as that in the eingenvalue/eigensystem-computation in the APT-article: it is just a simple procedure and works fast.
I was as well looking for a simpler, non-recursive description But I didn't find anything convincing. The only thing I found was that method which solves linear equation systems using the (u-1)(u^2-1)...() - denominators and assumes that at integer heights the numerator must be evenly divisible by that denominator plus the assumption, that the first and last row of the coefficients-matrices are the Stirlingnumbers first and second kind, respectively (or opposite). With this each coefficient-matrix can be determined separately, nonrecursive at least in the sense that the coefficients-matrices (and thus the computation of their second and their last columns) are independent of each other. (Don't know whether the description of this in the mentioned APT-article was too short/imprecise?) I didn't proceed there - it seemed to be too complicated and does not give an optimization over the efficient recursive procedure ; maybe you could find an improvement once the system of conditions is formally stated in terms of linear equations/matrix-formula and thus possibly get a more explicite expression in terms of the "curlybraces" ;-) only.
Well, I'll give it another try myself today, but my emphasis on this has a bit cooled down, to be frank...
Gottfried
Gottfried Helms, Kassel
