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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 30, 2012 1:35 PM

clicliclic@freenet.de wrote:
>
> Waldek Hebisch schrieb:

> >
> > clicliclic@freenet.de wrote:

> > >
> > > Waldek Hebisch schrieb:

> > > >
> > > > clicliclic@freenet.de wrote:

> > > > >
> > > > > In my view, the simple, trivial, answer as to why such formulae
> > > > > exist is that the "method of undetermined coefficients" works: In
> > > > > other words, one may insert polynomial factors and ramp up their
> > > > > degree until the linear system of equations that results after
> > > > > differentiation becomes soluble.

> > > >
> > > > Do you have a proof that this works? It seem that if you incresase
> > > > degree only on one side, then you end up with multiterm reccurences.
> > > > If you increase degree on both sides you have more equations.

> > >
> > > You are overlooking that there are three sides to the equation: the
> > > given integral, the transformed integral, and the integrated term. Extra
> > > "numerator" polynomials are inserted into each: the coefficients of that
> > > in the given integrand are assumed to be given, those of the other two
> > > assumed to be unknown. For a recurrence formula, the polynomials in the
> > > two integrands should have the same degree, and the highest admissible
> > > degree of that in the integrated term will obviously correlate with
> > > that. When the degrees are ramped up simultaneously, the numbers of
> > > unknowns rises twice as fast as the number of knowns, and a solution
> > > must eventually become feasible. This (fairly obvious) argument could no
> > > doubt be formalized into a mathematical proof, but I doubt the author
> > > would be interested in this.

> >
> > What is obvoius is that you do not have a proof -- you deal with
> > inhomogeneous system and merely counting number of equations
> > is not enough to proof consistency of such system.
> >
> > And there is quite trivial reason to be suspect of this argument.
> > Namely, it does not use assumptions about roots.

>
> I am not particularly interested in a formal proof of this (as opposed
> to being interested computer-algebra systems that produce continuous
> antiderivatives where possible, and can handle up to and including
> elliptic integrands), and the author of the paper doesn't look very
> interested either.

OK.

> Anybody who considers the existence of analogous formulae for higher
> degrees in doubt, should certainly try to provide a more formal proof.

The existence of formulas is not in doubt -- it is part of proof
of correctness of Hermite reduction.

FYI, Hermite reduction starts with an ansatz:

A*P_1^a_1*...*P_m^a_m = (B*P_1^(a_1+1)*...*P_m^(a_m+1))' +
C*P_1^(a_1+1)*...*P_k^(a_k+1)P_{k+1}^a_{k+1}*...*P_m^a_m

where P_i are relatively prime polynomials and A, B and C
are polynomials. Denoting by n_1 sum of degrees of P_i for i in 1..k
and by n sum of degrees of P_i for i in 1..m we require
A be polynomial of degree at most n-1, C be at most of degree n - 2,
while B at most of degree n_1 - 1.

A = B*S + B'*U + C*V

where U=P_1*...*P_m, S = ((a_1+1)P_1'/P_1 + ... + (a_m + 1)P_m'/P_m)*U,
V = P_1*...*P_k. Writing W = P_{k+1}*...*P_m, D = B'*W + C
we get

A = B*S + D*V

It is easy to see that as long as all (a_i + 1) for i=1,...,k are nonzero
S and V are relatively prime, so (by the extended euclidean algorithm)
there exist polynomials u and w such that

A = u*S + w*V

Moreover, when we requre u to be at most of degree n_1 - 1 = deg(V) - 1
the solution is unique, so B = u and D = w. Now C = w - B'*W and
it is easy to see that C satisfies requested degree bound.

This covers increasing a_i. For decreasing them argument is a bit
longer but not much.

My points are:

1) While argument is simple the solvability of resulting system
needs a proof.
2) The process is well-known as Hermite redution.
3) Since withing degree bounds solution is unique you must get
exactly the same formulas
4) At conceptual level it does not matter if you present this as
rules or an algorithm: you perform exactly the same sequence
of tranformations, using the same formulas. On practical level
procedures are likely to solve systems of equations on the fly,
while the paper presents precomputed formulas (and I agree that
precomputed formulas are better if you insist on encoding
Hermite reduction using rules).
5) Clearly Hermite reduction counts as prior art for the article.

> > > > > [...] Also, there is very little practical interest in going beyond
> > > > > elliptic integrals (or beyond Lauricella FD).

> > > >
> > > > Hmm, that is your opinion. Note however, that when integrating
> > > > functions of form R*sqrt(P) where R is a rational function and P is
> > > > of degree 3, the rules apply only if R is of degree 1. This seems to
> > > > be quite serious limitation. Of course you can split R into partial
> > > > fractions, but to get denominator which is power of linear factor may
> > > > require extra algebraic extension (in particular may require going to
> > > > complex numbers).

> > >
> > > Let's say, this summarizes my personal experience in science and
> > > engineering. I agree with the author that the x^1 and x^3 terms in an
> > > elliptic radicand are best annihilated right away to simplify the
> > > problem. If the result involves elliptic integrals of the third kind,
> > > Pi(x,n,k), the rational cofactor has to be split into partial fractions
> > > anyway in order to extract the parameters n, which may therefore come in
> > > complex conjugate pairs, etc.

> >
> > There are lucky cases where reduction process completely solves
> > the integral. In other cases, you may need algebraic extensions
> > for logarithmic and Pi terms, but algebraic part can be still
> > computed without extra extensions.
> >

>
> Agreed. But cases where this is absolutely crucial appear to be rare
> when problems relating to the physical world are addressed by
> integration.

Avoiding unneeded algebraics is crucial if you want to do some
symbolic computations with the result -- extra algebraics
can cause huge slowdown.

--
Waldek Hebisch
hebisch@math.uni.wroc.pl