>
> Well... My solution has problems,at list with 8.9.beta9 and also on
> Sagecell
> <https://sagecell.sagemath.org/?z=eJyFjzFrwzAQhXf9CuHQcpekJmnpVLR59OBkFUqRbTkIHMmcpWBa-t8r3LTuUOj44HvvvitF1-sQjAMpo9wptY5yr3jniUduHYfp9LyZTk-bPebkfRiBrDuLw6HWhApf-JBygKzkgmfbElklBvItyOkhLitlIm9gNYMVsqNuhLxqgoxe3z-yPMEXHSAiLj3S7mygT3YlovqeON4380haQFaI1jYB3uwAKSeDn0vFDBVJif5ySrTC3EyDdi0sevTlR8jugr2Y238V5Y03XWcba1Kcto-Yj7EeoUAhdshWK_4br73v_-vgJwyCdRQ=&lang=sage&interacts=eJyLjgUAARUAuQ==>,
>
> which is currently at 9.0) . Consider :
L=flatten([[u[0]]*u[1] for u in (x^5+x^3+1).roots(ring=QQbar)]); print("L =
",L)
P=prod([x-u for u in L]);print("P = ",P)
Rac=[var("r_{}".format(u)) for u in range(len(L))];print("R&c = ",Rac)
D=dict(zip(Rac,L));print("D = ",D)
Pr=prod([x-u for u in Rac]).expand();print("Pr = ",Pr)
%time print(Pr.coefficient(x,2).subs(D)==0)
But %time print(bool(Pr.coefficient(x,2).subs(D)==0)) never returns. Other
trials show that trying to use SR methods on expressions including QQbar
coeficients fails.
Is this known ? It seems to me that it's either a bug or a design flaw.
Advice ?
Le lundi 9 décembre 2019 12:32:42 UTC+1, Emmanuel Charpentier a écrit :
>
> Le lundi 9 décembre 2019 03:16:10 UTC+1, rjf a écrit :
>
> FYI:
>> In Maxima, integrate_use_rootsof: true
>> changes the result from an unevaluated integral to
>> lsum(log(x-%r1)/(4*%r1^3+2*%r1),%r1,rootsof(a+%r1^4+%r1^2,%r1))
>>
>> which is fairly compact, and to the point, but
>> which may or may not be what you want. The inside of rootsof
>> can be picked out and the 4 roots solved in terms of radicals
>> by Maxima's solve(). *
>>
> Aciddentally ! For example, [image: x^5+x^3+1] doesn’t have a solution in
> terms of radicals. Trying to solve it in Maxima fails ; and %solve gives
> a list of *numerical* roots.
>
> If the expression rootsof() is replaced by a list of the roots
>> returns by solve, { something like substpart( map(rhs, solve(
>> Z,3,1,x)), Z 3), lsum }
>> you get a large explicit mess of nested radicals and logs.
>>
> Indeed. And one may add that this solution has some drawbacks problems:
> computing only one value of the resultt primitive takes several seconds,
> trying to plot it is pointless…
>
> That Mathematical result with arctan is nicer that the explicit
>> radical stuff.
>>
> “Nicer” depends of what you want to do with the result. In some cases, the
> approximate answer obtained via %solve is preferable (e. g. plotting). In
> other cases, the explicit log/roots mess is preferable (e. g. trying to
> prove that the value of the primitive is real for all real values of the
> aurument).
>
> The “exact” answers given by Sage via (x^5+x^3+1).roots(ring=QQbar) may
> be preferable: a radical expression, if it exists, may be obtained via the
> radical_expression() method, and the precision is not limited to any
> specific floating-point approximation ; some equalities may be provable
> using them.
>
> OTOH, such values won’t pass to Maxima, and the resulting answers cannot
> be further simplified…
>
> The more general way to do this is :
>
> - remark that the polynoimial [image: x^5+x^3+1] has five (possibly
> confounded) roots, and name them [image: r_1,\dots,r_5] ;
> - rewrite it as [image: \displaystyle\prod_{r\in r_1,\dots,\r_5}(x-r)],
> - integrate using this rewriting,
> - substitute [image: r_1\dots,r_5] as needed by your various uses of
> the result.
>
> HTH,
>
>
>> RJF
>>
>>
>> On Friday, December 6, 2019 at 7:28:40 PM UTC-8, Emmanuel Charpentier
>> wrote:
>>>
>>> Okay. That means that we need
>>>
>>> - a wrapper for RootSum, AND
>>> - a wrapper for Lambda.
>>>
>>> ISTR that Mathematica has a similar setup.A rare occasion to kill two
>>> birds wit the same (two) stones. However, Mathematica's "pure functions"
>>> are, as far as I can tell, totally unknown to Sage...
>>>
>>> Thank you, Dima !
>>>
>>> Le mercredi 4 décembre 2019 23:21:39 UTC+1, Dima Pasechnik a écrit :
>>>>
>>>> On Wed, Dec 4, 2019 at 7:14 PM Emmanuel Charpentier
>>>> <emanuel.c...@gmail.com> wrote:
>>>> >
>>>> > The old (> 5 years...) Trac#16816 ticket is germane...
>>>> >
>>>> > BTW, I'm not sure that this would be very helpful:
>>>> >
>>>> > sage: foo=sympy.integrate(*[sympy.sympify(u) for u in (1/(x^4+x^2+a),
>>>> x)]); foo
>>>> > RootSum(_t**4*(256*a**3 - 128*a**2 + 16*a) + _t**2*(4 - 16*a) + 1,
>>>> Lambda(_t, _t*log(32*_t**3*a**2 - 8*_t**3*a + 4*_t*a - 2*_t + x)))
>>>> > sage: foo.doit()._sage_().factor().simplify_full()
>>>> > 1/4*sqrt(2)*(sqrt((4*a + sqrt(-64*a^3 + 48*a^2 - 12*a + 1) -
>>>> 1)/(16*a^3 - 8*a^2 + a))*log(1/4*sqrt(2)*((4*a^2 - a)*((4*a + sqrt(-64*a^3
>>>> + 48*a^2 - 12*a + 1) - 1)/(16*a^3 - 8*a^2 + a))^(3/2) + 2*sqrt(2)*x +
>>>> 2*(2*a - 1)*sqrt((4*a + sqrt(-64*a^3 + 48*a^2 - 12*a + 1) - 1)/(16*a^3 -
>>>> 8*a^2 + a)))) - sqrt((4*a + sqrt(-64*a^3 + 48*a^2 - 12*a + 1) - 1)/(16*a^3
>>>> - 8*a^2 + a))*log(-1/4*sqrt(2)*((4*a^2 - a)*((4*a + sqrt(-64*a^3 + 48*a^2
>>>> -
>>>> 12*a + 1) - 1)/(16*a^3 - 8*a^2 + a))^(3/2) - 2*sqrt(2)*x + 2*(2*a -
>>>> 1)*sqrt((4*a + sqrt(-64*a^3 + 48*a^2 - 12*a + 1) - 1)/(16*a^3 - 8*a^2 +
>>>> a)))) + sqrt((4*a - sqrt(-64*a^3 + 48*a^2 - 12*a + 1) - 1)/(16*a^3 - 8*a^2
>>>> + a))*log(1/4*sqrt(2)*((4*a^2 - a)*((4*a - sqrt(-64*a^3 + 48*a^2 - 12*a +
>>>> 1) - 1)/(16*a^3 - 8*a^2 + a))^(3/2) + 2*sqrt(2)*x + 2*(2*a - 1)*sqrt((4*a
>>>> -
>>>> sqrt(-64*a^3 + 48*a^2 - 12*a + 1) - 1)/(16*a^3 - 8*a^2 + a)))) - sqrt((4*a
>>>> - sqrt(-64*a^3 + 48*a^2 - 12*a + 1) - 1)/(16*a^3 - 8*a^2 +
>>>> a))*log(-1/4*sqrt(2)*((4*a^2 - a)*((4*a - sqrt(-64*a^3 + 48*a^2 - 12*a +
>>>> 1)
>>>> - 1)/(16*a^3 - 8*a^2 + a))^(3/2) - 2*sqrt(2)*x + 2*(2*a - 1)*sqrt((4*a -
>>>> sqrt(-64*a^3 + 48*a^2 - 12*a + 1) - 1)/(16*a^3 - 8*a^2 + a)))))
>>>> >
>>>> > Cheers ! ...
>>>> >
>>>> > However, Sage can come to the rescue. We can compute the root sum *in
>>>> Sage*, thus getting a "more interesting" result:
>>>> >
>>>> > sage: S1=sum([u[0]^u[1] for u in
>>>> foo.args[0]._sage_().roots()]).expand().simplify(); S1
>>>> > 1/2
>>>> >
>>>> > And (manually) use this result in the second expression. Here, I'm
>>>> stuck, because I do not (yet) understand how the lambda expression, second
>>>> argument of RooSum, is supposed to be used.
>>>>
>>>> RootSum(f(t), Lambda(y,g(y)) means
>>>>
>>>> sum_{y: f(y)=0} g(y)
>>>>
>>>> so we sum the function g() in Lambda over all the roots of the 1st
>>>> argument, f().
>>>>
>>>> >
>>>> > BTW, fricas can integrate this expression:
>>>> >
>>>> > sage:
>>>> IF=f(x).integrate(x,algorithm="fricas").expand().factor().simplify_full();
>>>> > ....: IF
>>>> > 1/4*sqrt(2)*(sqrt(((4*a^2 - a)*sqrt(-1/(4*a^3 - a^2)) + 1)/(4*a^2 -
>>>> a))*log(1/2*sqrt(2)*(2*sqrt(2)*x + ((4*a^2 - a)*sqrt(-1/(4*a^3 - a^2)) +
>>>> 4*a - 1)*sqrt(((4*a^2 - a)*sqrt(-1/(4*a^3 - a^2)) + 1)/(4*a^2 - a)))) -
>>>> sqrt(((4*a^2 - a)*sqrt(-1/(4*a^3 - a^2)) + 1)/(4*a^2 -
>>>> a))*log(1/2*sqrt(2)*(2*sqrt(2)*x - ((4*a^2 - a)*sqrt(-1/(4*a^3 - a^2)) +
>>>> 4*a - 1)*sqrt(((4*a^2 - a)*sqrt(-1/(4*a^3 - a^2)) + 1)/(4*a^2 - a)))) -
>>>> sqrt(-((4*a^2 - a)*sqrt(-1/(4*a^3 - a^2)) - 1)/(4*a^2 -
>>>> a))*log(1/2*sqrt(2)*(2*sqrt(2)*x + ((4*a^2 - a)*sqrt(-1/(4*a^3 - a^2)) -
>>>> 4*a + 1)*sqrt(-((4*a^2 - a)*sqrt(-1/(4*a^3 - a^2)) - 1)/(4*a^2 - a)))) +
>>>> sqrt(-((4*a^2 - a)*sqrt(-1/(4*a^3 - a^2)) - 1)/(4*a^2 -
>>>> a))*log(1/2*sqrt(2)*(2*sqrt(2)*x - ((4*a^2 - a)*sqrt(-1/(4*a^3 - a^2)) -
>>>> 4*a + 1)*sqrt(-((4*a^2 - a)*sqrt(-1/(4*a^3 - a^2)) - 1)/(4*a^2 - a)))))
>>>> >
>>>> > And fiddling with sympy.cse suggests that the reference given might
>>>> be correct:
>>>> >
>>>> > sage: sympy.cse(sympy.sympify(IF), optimizations="basic")
>>>> > ([(x0, sqrt(2)),
>>>> > (x1, 4*a),
>>>> > (x2, x1 - 1),
>>>> > (x3, 1/x2),
>>>> > (x4, a*x2*sqrt(-x3/a**2)),
>>>> > (x5, x4 + 1),
>>>> > (x6, x3/a),
>>>> > (x7, sqrt(x5*x6)),
>>>> > (x8, 2*x*x0),
>>>> > (x9, x7*(x2 + x4)),
>>>> > (x10, x0/2),
>>>> > (x11, sqrt(-x6*(x4 - 1))),
>>>> > (x12, x11*(-x1 + x5))],
>>>> > [-x0*(-x11*log(x10*(-x12 + x8)) + x11*log(x10*(x12 + x8)) +
>>>> x7*log(x10*(x8 - x9)) - x7*log(x10*(x8 + x9)))/4])
>>>> >
>>>> > HTH,
>>>> >
>>>> > Le mercredi 4 décembre 2019 16:38:34 UTC+1, Eric Gourgoulhon a écrit
>>>> :
>>>> >>
>>>> >> Hi,
>>>> >>
>>>> >> In Sage 9.0.beta8 we have
>>>> >>
>>>> >> sage: a = var('a')
>>>> >> sage: integrate(1/(x^4 + x^2 + a), x)
>>>> >> ...
>>>> >> AttributeError: 'RootSum' object has no attribute '_sage_'
>>>> >>
>>>> >> The same error occurs in Sage 8.9, but not in Sage 8.8 (and below).
>>>> In Sage 8.8, we have instead:
>>>> >>
>>>> >> sage: a = var('a')
>>>> >> sage: integrate(1/(x^4 + x^2 + a), x)
>>>> >> integrate(1/(x^4 + x^2 + a), x)
>>>> >>
>>>> >> Note that RootSum is a SymPy object. Actually, in Sage 9.0.beta8,
>>>> forcing the algorithm to 'maxima' yields the same result as in Sage 8.8:
>>>> >>
>>>> >> sage: integrate(1/(x^4 + x^2 + a), x, algorithm='maxima')
>>>> >> integrate(1/(x^4 + x^2 + a), x)
>>>> >>
>>>> >> So it seems that since Sage 8.9, when integrate() is not capable to
>>>> find an answer via Maxima, it tries SymPy but is not capable to translate
>>>> the result back to Sage. I could not find a ticket about this. Shall I
>>>> open
>>>> one?
>>>> >>
>>>> >> Eric.
>>>> >>
>>>> >> PS: for the record, a primitive of 1/(x^4 + x^2 + a) is
>>>> >>
>>>> >> sage: b = sqrt(1 - 4*a)
>>>> >> sage: f = sqrt(2)/b*(arctan(sqrt(2)*x/sqrt(1 - b))/sqrt(1 - b) -
>>>> arctan(sqrt(2)*x/sqrt(1 + b))/sqrt(1 + b))
>>>> >>
>>>> >> as we can check:
>>>> >>
>>>> >> sage: diff(f, x).simplify_full()
>>>> >> 1/(x^4 + x^2 + a)
>>>> >>
>>>> >>
>>>> > --
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>>>>
>>>>
>>>>
>>>
>
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