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Topic: An independent integration test suite
Replies: 128   Last Post: Dec 8, 2013 3:21 PM

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 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: The A. F. Timofeev symbolic integration test suite
Posted: Nov 21, 2013 2:14 PM

Albert Rich schrieb:
>
> On Saturday, November 16, 2013 2:47:27 AM UTC-10, clicl...@freenet.de wrote:
>

> > Would the evaluations of Examples 3.n, 5a.n, 5b.n and 17 perhaps also
> > profit from Euler's transformation?

>
> Yes, I think Euler would approve of the optimal antiderivatives now
> shown for examples 3.n, 5a.n, 5b.n, 6a.n, 6b.n, 14 and 17 in the
> Timofeev Chapter 8 pdf file at
>
> http://www.apmaths.uwo.ca/~arich/TimofeevChapter8TestResults.pdf
>

> > I have rummaged my vaults and dug up a mildewed sheaf of papers with the
> > gospel on 2F1 differentiation and integration. The integration part
> > reads as follows (typed in without checking):
> >
> > INT(F21(a,b,c,x), x)
> > = (c-1)/((a-1)*(b-1))*F21(a-1,b-1,c-1,x)
> > [a/=1, b/=1, c/=1]
> >
> > INT(x^(b-2)*F21(a,b,c,x), x)
> > = 1/(b-1)*x^(b-1)*F21(a,b-1,c,x)
> > [b/=1]
> >
> > INT(x^(c-1)*F21(a,b,c,x), x)
> > = 1/c*x^c*F21(a,b,c+1,x)
> >
> > INT((1-x)^(b-2)*F21(a,b,c,x), x)
> > = (c-1)/((a-c+1)*(b-1))*(1-x)^(b-1)*F21(a,b-1,c-1,x)
> > [b/=1, c-a/=1, c/=1]
> >
> > INT(x^(c-1)*(1-x)^(b-c-1)*F21(a,b,c,x), x)
> > = 1/c*x^c*(1-x)^(b-c)*F21(a+1,b,c+1,x)
> >
> > This set should be augmented by applying Euler's transformation on both
> > sides of each formula. Inasmuch as the 2F1 integration rules are unknown
> > to Rubi, I suggest to implement the complete set. [...]

>
> Rubi is an open-source project that needs contributions by others than
> me in order to reach its full potential. Also I am not an expert in
> hypergeometric functions and have no desire to become one. However, I
> would be delighted to incorporate a hypergeometric integration package
> written by someone knowledgeable in the field, like yourself...
>

Should be no problem once Rubi can be run without Mathematica (Warning:
when I work on something like this, it is likely to accumulate changes
everywhere). Actually, the more special a function, the fewer its
integration rules; so what remains to be done on cylinder (Bessel) and
parabolic cylinder functions, and on confluent and generalized
hypergeometric and Meijer-G functions is pretty limited; Jacobian
elliptic functions and friends may be the hardest part (cf.
Labahn-Humphries 2005). Possibilities multiply, however, when one
restricts to definite integrals over special intervals, such as the
positive real semiaxis - but this is a separate story.

> > Looking forward to Rubi4.3forte,
>
> Although not formally announced, Rubi 4.3 is now available for
>
> http://www.apmaths.uwo.ca/~arich/
>

... Soul clap hands and sing, and louder sing ...

> In addition to Euler's transformation it includes numerous
> improvements including the use of rectification to produce continuous
> antiderivatives after integrating trig expressions using the
> substitution u=tan(x) or u=tan(x/2). The algorithm is described in
> D.J.Jeffrey's 1997 paper "Rectifying Transformations for the
> Integration of Rational Trigonometric Functions" available at
>
> http://www.apmaths.uwo.ca/~djeffrey/Offprints/trig-rec.pdf
>

I haven't taken a close look yet but hope this is not what you called a
"messy algorithm [] not appropriate for an elegant, rule-based system;
[] it requires the host CAS provide a strong limit package and good
algebraic simplification." Such requirements would severely limit the
portability of Rubi.

Martin.

Date Subject Author
2/24/13 clicliclic@freenet.de
3/19/13 clicliclic@freenet.de
3/21/13 Waldek Hebisch
3/22/13 clicliclic@freenet.de
3/26/13 Waldek Hebisch
3/26/13 clicliclic@freenet.de
4/20/13 clicliclic@freenet.de
4/20/13 Nasser Abbasi
4/20/13 Rouben Rostamian
4/20/13 clicliclic@freenet.de
4/20/13 Rouben Rostamian
4/20/13 Axel Vogt
4/20/13 clicliclic@freenet.de
4/20/13 Axel Vogt
4/21/13 Axel Vogt
4/21/13 clicliclic@freenet.de
4/21/13 Waldek Hebisch
4/22/13 clicliclic@freenet.de
4/22/13 Axel Vogt
4/22/13 clicliclic@freenet.de
4/23/13 Waldek Hebisch
4/24/13 clicliclic@freenet.de
4/25/13 Waldek Hebisch
4/26/13 clicliclic@freenet.de
4/27/13 Waldek Hebisch
4/24/13 Richard Fateman
4/24/13 clicliclic@freenet.de
4/25/13 Richard Fateman
4/26/13 clicliclic@freenet.de
4/26/13 Axel Vogt
4/27/13 clicliclic@freenet.de
4/25/13 Waldek Hebisch
4/25/13 Peter Pein
4/25/13 Nasser Abbasi
4/26/13 Peter Pein
4/26/13 clicliclic@freenet.de
4/26/13 Peter Pein
4/26/13 clicliclic@freenet.de
4/26/13 Richard Fateman
4/27/13 clicliclic@freenet.de
4/27/13 Richard Fateman
6/30/13 clicliclic@freenet.de
6/30/13 Axel Vogt
7/1/13 clicliclic@freenet.de
7/1/13 Axel Vogt
7/1/13 Waldek Hebisch
7/2/13 clicliclic@freenet.de
7/2/13 clicliclic@freenet.de
7/2/13 clicliclic@freenet.de
7/2/13 Nasser Abbasi
7/2/13 Nasser Abbasi
7/4/13 clicliclic@freenet.de
7/4/13 Nasser Abbasi
7/4/13 Nasser Abbasi
7/5/13 clicliclic@freenet.de
7/5/13 Nasser Abbasi
7/9/13 clicliclic@freenet.de
7/10/13 Nasser Abbasi
7/10/13 Richard Fateman
7/10/13 Nasser Abbasi
7/10/13 clicliclic@freenet.de
8/6/13 clicliclic@freenet.de
9/15/13 Albert D. Rich
9/15/13 clicliclic@freenet.de
9/15/13 clicliclic@freenet.de
9/21/13 Albert D. Rich
9/21/13 clicliclic@freenet.de
9/22/13 daly@axiom-developer.org
9/24/13 daly@axiom-developer.org
9/30/13 daly@axiom-developer.org
9/22/13 Albert D. Rich
9/25/13 Albert D. Rich
9/25/13 Albert D. Rich
9/25/13 clicliclic@freenet.de
9/25/13 Albert D. Rich
9/26/13 Albert D. Rich
9/26/13 clicliclic@freenet.de
9/26/13 Albert D. Rich
9/29/13 clicliclic@freenet.de
10/1/13 Albert D. Rich
10/1/13 clicliclic@freenet.de
10/1/13 Albert D. Rich
10/5/13 clicliclic@freenet.de
10/5/13 Albert D. Rich
10/6/13 clicliclic@freenet.de
10/10/13 Albert D. Rich
10/10/13 Nasser Abbasi
10/11/13 clicliclic@freenet.de
11/6/13 Albert D. Rich
11/6/13 Nasser Abbasi
11/7/13 did
11/7/13 clicliclic@freenet.de
11/7/13 clicliclic@freenet.de
11/7/13 Albert D. Rich
11/12/13 clicliclic@freenet.de
11/12/13 Albert D. Rich
11/13/13 clicliclic@freenet.de
11/13/13 Albert D. Rich
11/14/13 clicliclic@freenet.de
11/14/13 Albert D. Rich
11/15/13 clicliclic@freenet.de
11/15/13 Albert D. Rich
11/16/13 clicliclic@freenet.de
11/16/13 clicliclic@freenet.de
11/21/13 Albert D. Rich
11/21/13 clicliclic@freenet.de
11/21/13 Nasser Abbasi
11/21/13 Albert D. Rich
11/21/13 Albert D. Rich
11/22/13 clicliclic@freenet.de
11/14/13 Albert D. Rich
11/15/13 clicliclic@freenet.de
11/15/13 Nasser Abbasi
11/16/13 clicliclic@freenet.de
11/16/13 Nasser Abbasi
11/7/13 did
11/7/13 clicliclic@freenet.de
4/20/13 Richard Fateman
4/21/13 clicliclic@freenet.de
4/20/13 Axel Vogt
4/20/13 clicliclic@freenet.de
4/20/13 Waldek Hebisch
4/21/13 G. A. Edgar
12/8/13 clicliclic@freenet.de
10/5/13 Albert D. Rich
10/6/13 clicliclic@freenet.de