After last week’s breakthrough, work this week has been very slow. I started working on the Parametric Risch Differential Equation Problem, which is almost identical to the Risch Differential Equation Problem in how it is solved, except there are a few extra steps. Unfortunately, because it is so similar, Bronstein breezes through the description. This is fine for the parts that are the same, but he is a little unclear on how some of the new parts fit in. Also, his pseudocode has a line more or less saying
if r1 = … = rn = 0 then N = -1 else N = max(deg(r1), …, deg(rn)) for i from 0 to N for j from 1 to m Mij = coefficient(rj, t^i)
where M is a matrix. It is not very clear what this is supposed to mean in the case where N = -1. Obviously, you can’t have a a matrix with negative dimensions. Clearly, this means that this particular function doesn’t apply somehow in this case, but I am not really even sure where it fits in to the whole algorithm at this point in reading. After reading a few more pages in, it gives a few hints here and there on how it is to be used, but never is it explicitly shown, in pseudocode or otherwise. So for now, I think my best bet is to read ahead and get a fuller understanding of the complete function before I try implementing anything (this is what I had been doing before, but I caught up to myself).
Also, on an unrelated note, I just found out today that I passed my Google Summer of Code midterm evaluation. This means that I will receive half of my stipend for the program (the other half comes after passing the final evaluation at the end of the summer), and that I can continue working on my project in the program.
Later in the text, it runs through an example and says “… , hence M and A are 0 by 0 matrices.” So obviously, that is what was meant.