08/14/2007, 02:42 AM
It seems clear to me that these polynomials should be stored in a matrix. You have two sets of variables: n^0, n^1, n^2, ..., and z^0, z^1, z^2, ..., and a you have one coefficient for each combination, so long as the degree of n is less than the degree of z.
Depending on whether you make the z's a row vector or a column vector, you'll either have an upper triangular matrix or a lower triangular matrix. Once in that format, it should be much easier to pick out the patterns, and it should provide an effective viewpoint to assess convergence.
At any rate, I'm not terribly concerned if the radius of convergence is goes to 0 as the number of terms goes to infinity. There should still be a limit that is well-behaved, even if it means taking the limit as z goes to 0, with an integer iteration count that goes to infinity to get us back up the vicinity of z=1.
This is essentially what I did with my cheta function, but with linear interpolation. With a higher degree interpolation function based on the power series derived here (e.g., the first 20 terms should converge very nicely for z <0.01), convergence should be even more well-behaved, and by extension, the limit that much more defensible. Assuming the limit is defensible, then less accuracy could be acceptable by using a non-limited approximation.
And this is assuming the radius of convergence indeed goes to 0. I don't have access to Baker's proof, so I don't have a lot to go on at the moment.
Depending on whether you make the z's a row vector or a column vector, you'll either have an upper triangular matrix or a lower triangular matrix. Once in that format, it should be much easier to pick out the patterns, and it should provide an effective viewpoint to assess convergence.
At any rate, I'm not terribly concerned if the radius of convergence is goes to 0 as the number of terms goes to infinity. There should still be a limit that is well-behaved, even if it means taking the limit as z goes to 0, with an integer iteration count that goes to infinity to get us back up the vicinity of z=1.
This is essentially what I did with my cheta function, but with linear interpolation. With a higher degree interpolation function based on the power series derived here (e.g., the first 20 terms should converge very nicely for z <0.01), convergence should be even more well-behaved, and by extension, the limit that much more defensible. Assuming the limit is defensible, then less accuracy could be acceptable by using a non-limited approximation.
And this is assuming the radius of convergence indeed goes to 0. I don't have access to Baker's proof, so I don't have a lot to go on at the moment.
~ Jay Daniel Fox
