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)
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
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.
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)
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
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.
cliclic...@freenet.de wrote: >... > they claim to deliver the antiderivative of any elementary function if > it can also be written in terms of elementary functions...
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.
> cliclic...@freenet.de wrote: > >... > > they claim to deliver the antiderivative of any elementary function if > > it can also be written in terms of elementary functions...
> 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.
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?
> 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)
> 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
> 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.
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!
> > cliclic...@freenet.de wrote: > > >... > > > they claim to deliver the antiderivative of any elementary function if > > > it can also be written in terms of elementary functions...
> > 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.
> 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 ;)
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
> On Oct 19, 5:16 pm, cliclic...@freenet.de wrote: > > Richard Fateman schrieb:
> > > cliclic...@freenet.de wrote: > > > >... > > > > they claim to deliver the antiderivative of any elementary function if > > > > it can also be written in terms of elementary functions...
> > > 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.
> > 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 ;)
> Interesting. Where do you find such patterns? > Do you have a catalog of them I can try?
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"?
> > On Oct 19, 5:16 pm, cliclic...@freenet.de wrote: > > > Richard Fateman schrieb:
> > > > cliclic...@freenet.de wrote: > > > > >... > > > > > they claim to deliver the antiderivative of any elementary function if > > > > > it can also be written in terms of elementary functions...
> > > > 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.
> > > 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 ;)
> > Interesting. Where do you find such patterns? > > Do you have a catalog of them I can try?
> 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.
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.
> 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.
