firstname.lastname@example.org wrote: <snipped quotation only post> > Can I be wrong in suspecting the evil hand of Professor Moriarty here? >
Below is what I intended to post:
email@example.com wrote: > > Hooray, hooray. A paper has finally appeared on two-term recurrence > formulae for indefinite algebraic integrals: > > <http://arxiv.org/abs/1209.3758> > > I wonder though if this might be a hoax.
Does not look as a hoax :). But AFAICS there is one (maybe 2-3 depending on how you count) theorem, which author did not state and a lot of examples. More precisely, author wrote:
> The two-term recurrence relations have been derived by the method > of undetermined coefficients
Of course the interesting question is why such formulas should exist. The answer (which the author apparently did not want to disclose) is that Hermite reduction method works. Fact that it works for increasing exponents by 1 is well-known. Fact that it can be used to reduce exponents by 1 is less known, but for example Bronstein mentions this in his thesis.
Author also did not mention easy to observe fact: given
where A is product of roots (with possibly added exponential factor), Q is product of powers of polynomials Q_1, ..., Q_n containing all radicands, such that sum of degrees of Q_1, ..., Q_n is m and R is a polynomial of degree at most m - 1, one can subtract multiple of (Q*A)' from R*Q*A and get a similar term with R of degree at most m - 2. So using the R term of degree m - 2 he still effectively has the same coverage as Hermite reduction.
The author repeatedy writes phrases like:
> To exclude integrands with confluent roots, the following > recurrences should be applied only if the resultant of the > linear polynomials does not vanish
I do not know why he wants to exclude confluent roots, because AFAICS the formulas are equivalent to equalites between polynomials, so are valid for all values of parameters. When applying them we need to avoid division by zero, which in general is different condition than excluding confluent roots.
Also, I find his motivation form introdution somewhat disconnected with rest of the article. Namely, Hermite reduction seem to widely used and does not eliminate form of integrals that the author does not like. AFAICS the main source of difficulty is due to logarithmic terms, which is outside of Hermite reduction. Minor source of difficulty is because some (otherwise attractive) simplifications can change branching pattern of the integral. Hermite reduction is of limited relevance for the seond problem - it gives "rational" approach which works without additional simplifications, but simplifications are typically introduced because of other steps. Even in contexts of rule based integration it may be better to keep Hermite reduction as a procedure istead of encoding it as set of rules.
The author precomputes results of Hermite reduction for a few "typical" forms of integrand. If this is worth the effort can be decided only for an integrator as a whole, but probably in some cases precomputed formulas give large saving in compute time.