08/25/2009, 09:43 PM
Henryk, I've found some rather bizarre behvior with your method of shifting the Abel function's center. For example, it seems fairly well behaved for real shifts, but even a small imaginary shift seems to produce garbage results (i.e., as I increase the matrix size, I don't get convergence, but rather rapid divergence).
I can use complex math with f(x)=e*(x+x0)-x0 to get the power series for log(x+x0), except the leading constant term of course. This holds true for each value of x0 I have tried so far, including complex values or pure imaginary values.
However, I was unable to successfully recenter the islog_e to x=0.25*i, which is a very small shift, relatively speaking. Why should we be unable to shift away from the real axis, if we are not limited by a radius of convergence?
Actually, I've done a bit more testing just now, and it seems that the answer works if I use e^x-x0, and rather than subtract the identity matrix, i.e., the Carleman matrix of f(x)=x, I instead subtract the Carleman matrix of f(x)=x-x0.
Now that I look at this, I see that your method and mine are essentially identical, except that you have an additional shift involved. I use the Carleman matrices for f(x) = e^(x)-x0 and g(x) = x-x0, while you use the Carleman matrices for f(x) = e^(x+x0)-x0 and g(x) = (x+x0) - x0. In other words, you have this additional substitution of x = x+x0. (Alternatively, I have an extra substitution relative to your method!) This additional shift doesn't seem to be a problem for real-valued shifts, but it's a huge problem for complex shifts.
Oddly enough, where my method is unable to get safely outside the radius of convergence, yours does so readily, though for reals only.
I can use complex math with f(x)=e*(x+x0)-x0 to get the power series for log(x+x0), except the leading constant term of course. This holds true for each value of x0 I have tried so far, including complex values or pure imaginary values.
However, I was unable to successfully recenter the islog_e to x=0.25*i, which is a very small shift, relatively speaking. Why should we be unable to shift away from the real axis, if we are not limited by a radius of convergence?
Actually, I've done a bit more testing just now, and it seems that the answer works if I use e^x-x0, and rather than subtract the identity matrix, i.e., the Carleman matrix of f(x)=x, I instead subtract the Carleman matrix of f(x)=x-x0.
Now that I look at this, I see that your method and mine are essentially identical, except that you have an additional shift involved. I use the Carleman matrices for f(x) = e^(x)-x0 and g(x) = x-x0, while you use the Carleman matrices for f(x) = e^(x+x0)-x0 and g(x) = (x+x0) - x0. In other words, you have this additional substitution of x = x+x0. (Alternatively, I have an extra substitution relative to your method!) This additional shift doesn't seem to be a problem for real-valued shifts, but it's a huge problem for complex shifts.
Oddly enough, where my method is unable to get safely outside the radius of convergence, yours does so readily, though for reals only.
~ Jay Daniel Fox