Is your CAS no a weakling anymore?
clicl...@freenet.de
Hello,
last year's October saw a demonstration that large commercial and free
Computer Algebra systems like Math-o'-Maple and Maxioma are weaklings:
they claim to deliver the antiderivative of any elementary function if
it can also be written in terms of elementary functions (the magic
algorithms behind this are too arcane to disclose, of course), but they
all flunked on Martin's elementary double integral:
int(int(jxx(x-a/2, y, z) * jxx(x+a/2, y, z) + jxy(x-a/2, y, z) *
jxy(x+a/2, y, z), x, -inf, inf), y, -inf, inf)
where
jxx(x,y,z) := -x*y*(2*(x^2+y^2+z^2)^(3/2) - 2*z^3 - 3*z*(x^2+y^2)) /
((x^2+y^2+z^2)^(3/2)*(x^2+y^2)^2)
jxy(x,y,z) := ((x^2-y^2)*(x^2+y^2+z^2)*((x^2+y^2+z^2)^(1/2) - z) -
z*x^2*(x^2+y^2)) / ((x^2+y^2+z^2)^(3/2)*(x^2+y^2)^2)
and z must be restricted to positive values: z >= 0.
For the simple transformations of this problem leading to its solution
via elementary antiderivatives on Derive 6.10 see last year's thread
under
<http://groups.google.com/group/sci.math.symbolic/browse_thread/thread/c42b461e9c9710e3/f3ea3864b161b989?lnk=raot&fwc=2>.
Now, here is a chance for CAS developers and their customers to check on
the progress made since last year, and to hone their skills as CAS
pilots. What is the simple, elementary result of
int(int(jxx(x, y-a/2, z) * jxx(x, y+a/2, z) + jxy(x, y-a/2, z) *
jxy(x, y+a/2, z), x, -inf, inf), y, -inf, inf)
where again z >= 0 ?
If you can tease the answer out of your CAS, please show us the major
steps towards it!
Martin.
PS: Remember not to pay USD 5000 next time unless your CAS vendor can
demonstrate the usefulness of his integrator with these nice elementary
double integrals.
clicl...@freenet.de
Richard Fateman
>...
> they claim to deliver the antiderivative of any elementary function if
At least Maxima / Macsyma makes no such claims, and in particular
neither implements completely the algebraic case of the Risch algorithm.
Undoubtedly simpler examples will stump these programs.
clicl...@freenet.de
Richard Fateman schrieb:
Errh. Ok. Alright. Maxima users are excused for the time being - they
may hand in their answers after the next integrator overhaul. On the
other hand, Derive also makes do with less than 1000 general
integration rules like
INT(F((a+b*x)^(1/n),x),x) ->
n/b*SUBST(INT(x^(n-1)*F(x,(x^n-a)/b),x),x,(a+b*x)^(1/n)),
or
INT(x^m*LN((a*x^n)^q),x) ->
x^(m+1)*LN((a*x^n)^q)/(m+1)-n*q*x^(m+1)/(m+1)^2.
What counts in the end is the ability to handle real-life integrals like
the problem posed. Maybe Risch's is not the best way?
And now pssss. They are all crouched over their screens. No sound but
the occasional keyboard click and slurp of coffee. Papa Wolfram looks
particularly grave. Will he and his crew flunk again?
Martin ;)
clicl...@freenet.de
clicl...@freenet.de schrieb:
There is a tendency of Computer Algebra systems to embark on
calculations that take an unreasonable time, or to produce expressions
that are needlessly large - sometimes too huge to be processed further
or even verified. Such behavior is probably believed to sell better than
simply returning with no result. The real-life integration problem posed
above demonstrates this tendency, and may thereby help eliminate it.
Four weeks have now passed since the problem was posed. There is no
coffee left, and Papa Wolfram instantly goes into trance when his eyes
come to rest on a logarithm of an algebraic function. But no solution,
or hint of a solution, on the Computer Algebra systems Axiom, Maple, or
Mathematica has been shown. It may be concluded that the performance of
these systems (or perhaps just the skill of their operators) with
respect to this kind of elementary double integral remains as
unsatisfactory as it was one year ago.
However, as not to spoil the fun of those die-hard's still working on
the problem, the solution will not be disclosed here and now: if you
would like to have a final result or intermediate results for checking
purposes, please refer to last year's closely related integral and its
solution on Derive.
Once a weakling, always a weakling? Or: What doesn't kill us makes us
stronger?
Martin.
PS: Note that the kind of public noise made in this thread was seen to
be effective before: Since 2008 there finally is a Mathematica function
implementing the entire Lerch transcendent rather than merely half of
it!
TimDaly
> Richard Fateman schrieb:
Interesting. Where do you find such patterns?
Do you have a catalog of them I can try?
Tim Daly
Axiom Lead Developer
Elder of the Internet
clicl...@freenet.de
TimDaly schrieb:
Could you be more specific? Does your "catalog of patterns" refer to my
integration problems, or perhaps to my associative memory connecting
attempts to solve them with concepts like "coffee" and "Wolfram" (which
is further linked to "Papa") and "trance"?
Martin.
TimDaly
Do you have a catalog of patterns (in source code) of the form:
INT(F((a+b*x)^(1/n),x),x) ->
n/b*SUBST(INT(x^(n-1)*F(x,(x^n-a)/b),x),x,(a+b*x)^(1/n)),
I would like to see what patterns you have and compare them against
the integration test suite that Axiom uses.
Tim
clicl...@freenet.de
TimDaly schrieb:
>
> Do you have a catalog of patterns (in source code) of the form:
> INT(F((a+b*x)^(1/n),x),x) ->
> n/b*SUBST(INT(x^(n-1)*F(x,(x^n-a)/b),x),x,(a+b*x)^(1/n)),
>
> I would like to see what patterns you have and compare them against
> the integration test suite that Axiom uses.
>
I see. Yes, I have these rule strings for Derive 6.10 in the form shown
above. The integrator in particular employs 611 transformation rules
amounting to 60k of ASCII text. In single-step mode, the strings are
displayed by Derive along with intermediate results. I know these rules
only to the extent that the corresponding strings are present in the
Derive executable; if rules are applied that are not accompanied by a
string, they would be missing in my set (but so far I haven't noticed
gaps of this kind in the integrator).
I suspect that copyright doesn't allow me or you to make the full set of
Derive's integrator rules public. If you declare your agreement with
this restriction, I will e-mail the full set to you, however. Please
tell me what address to send them to.
Martin.