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Topic: 136 theorems on 29 pages
Replies: 20   Last Post: Nov 19, 2012 4:55 PM

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Waldek Hebisch

Posts: 267
Registered: 12/8/04
Re: 136 theorems on 29 pages
Posted: Sep 25, 2012 9:17 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply 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: wrote:
> Hooray, hooray. A paper has finally appeared on two-term recurrence
> formulae for indefinite algebraic integrals:
> <>
> 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.

Waldek Hebisch

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