Re: [sage-support] Matrix logarithm
On Mon, 18 Apr 2022 at 14:30, GUSTAVO TERRA BASTOS wrote:
>
> Hi guys.
>
> Given two n x n matrices M, N, we know it is a big problem to find the
> positive integer "i" such that M^i = N (There are other hypothesis involved).
> In my particular case, I would like to do the same for 3 x 3 matrices M , N
> over F_{11^2} (finite field with 121 elements).
Note that finding this integer i here is something different from what
is usually referred to as the "matrix logarithm" (the inverse of the
matrix exponential) which gives a matrix rather than an integer as the
result.
Is it prohibitive to just compute powers of M until you find that M^i = N?
Any i would need to satisfy det(M)^i = det(N) which should be faster
to solve for large i unless det(M) and det(N) are both zero or one.
Otherwise I think you can generalise the approach to other
coefficients in the characteristic polynomial.
Oscar
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Re: [sage-support] Plot doesn't respect numerical precision specification.
On Tue, 6 Dec 2022 at 12:53, Emmanuel Charpentier wrote: > > > And Sympy currently never returns. With current SymPy master it takes 6 seconds: In [1]: %time log(tan(pi/2*tanh(x))).diff(x).limit(x, oo) CPU times: user 5.73 s, sys: 24 ms, total: 5.75 s Wall time: 5.75 s Out[1]: 2 The fix was part of a GSOC project this summer and will be in SymPy 1.12: https://github.com/sympy/sympy/pull/23844 -- Oscar -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/CAHVvXxSvaiH1mKt4kNgfs%2Bndhf22B2Hsvyav0tTHwpFC6Oq9Uw%40mail.gmail.com.
Re: [sage-support] Integral result differ from Wolfram|Alpha
On Mon, 13 Nov 2023 at 21:32, Bùi Gia Nghĩa wrote: > > Hi! > I have used Sage Cell Server to integrate the function (ln(x)^2 - 1) / (x * > ln(x)). It should resulted in (ln(x)^2) / 2 - |ln(ln(x))| + C, as noted by my > textbook and Wolfram|Alpha, but instead resulted in 1/2*log(x)^2 - > 1/2*log(log(x)^2). Your textbook assumes that you are only interested in real antiderivatives and so describes the antiderivative with abs that is not complex-differentiable (i.e. not valid for non-real values of x). The result from Sage is valid for e.g. x = 1+i whereas your textbook result would be incorrect in that case. -- Oscar -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/CAHVvXxTrpcPGrhTnfVfNkKvyf-dJBr-BCzT_NEqVonKRDEt8bQ%40mail.gmail.com.
Re: [sage-support] Re: limitations of "solve"?
On Tue, 28 Nov 2023 at 17:25, kcrisman wrote: > > Answering part of my question: it seems that sympy and maxima have > different attitudes towards fractional powers of negative numbers, which > may be the source of the problem. > > Yes. Maxima's attitude is that the square root of negative one is an > expression which might have multiple values, rather than just picking one you > hope might be consistent over branch points. (There are very long > discussions on this list about this from years ago, which I will spare you by > not linking to any of them.) I would not be surprised if that came up here. I wouldn't mind seeing those discussions if you have a link to send (perhaps only to me). -- Oscar -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/CAHVvXxQKe1HvcD%2BYL3CHTpyH7s2K_LeEEq6Z-wJKjAmYjw5jGQ%40mail.gmail.com.
Re: [sage-support] Re: limitations of "solve"?
On Wed, 29 Nov 2023 at 12:40, Eric Gourgoulhon wrote:
>
> Le mardi 28 novembre 2023 à 18:25:04 UTC+1, kcrisman a écrit :
>
> Yes. Maxima's attitude is that the square root of negative one is an
> expression which might have multiple values, rather than just picking one you
> hope might be consistent over branch points.
>
> To enforce Maxima to work in the real domain, avoiding to play too much with
> complex square roots, one can add at the beginning of the Sage session:
>
> maxima_calculus.eval("domain: real;")
>
> Then the second example in the initial message of this thread yields
>
> [[x == 2/5*sqrt(6)*sqrt(5), y == 16, l == 1/9*18750^(1/6)], [x ==
> -2/5*sqrt(6)*sqrt(5), y == 16, l == -1/9*18750^(1/6)]]
>
> instead of an empty list.
When using Maxima (5.45.1) directly I get this result with default settings:
(%i1) f: 10*x^(1/3)*y^(2/3)$
(%i2) g: 5*x^2 + 6*y$
(%i3) solve([diff(f,x)=l*diff(g,x), diff(f,y)=l*diff(g,y), g=120], [x,y,l]);
1/6
2 sqrt(6) 18750
(%o3) [[x = -, y = 16, l = ],
sqrt(5) 9
1/6
2 sqrt(6)18750
[x = - -, y = 16, l = - ]]
sqrt(5)9
Does Sage modify some Maxima settings related to this or does it call
something other than solve?
--
Oscar
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Re: [sage-support] Re: limitations of "solve"?
What does "set domain to complex" mean in terms of Maxima's settings?
Maxima's solve seems to compute complex solutions by default:
(%i21) solve(x^2 + 1);
(%o21)[x = - %i, x = %i]
On Sun, 3 Dec 2023 at 13:37, Dima Pasechnik wrote:
>
> Yes, Sage modifies the defaults of Maxima, in particular we set domain to
> complex.
>
> On 3 December 2023 12:28:45 GMT, Oscar Benjamin
> wrote:
> >On Wed, 29 Nov 2023 at 12:40, Eric Gourgoulhon
> >wrote:
> >>
> >> Le mardi 28 novembre 2023 à 18:25:04 UTC+1, kcrisman a écrit :
> >>
> >> Yes. Maxima's attitude is that the square root of negative one is an
> >> expression which might have multiple values, rather than just picking one
> >> you hope might be consistent over branch points.
> >>
> >> To enforce Maxima to work in the real domain, avoiding to play too much
> >> with complex square roots, one can add at the beginning of the Sage
> >> session:
> >>
> >> maxima_calculus.eval("domain: real;")
> >>
> >> Then the second example in the initial message of this thread yields
> >>
> >> [[x == 2/5*sqrt(6)*sqrt(5), y == 16, l == 1/9*18750^(1/6)], [x ==
> >> -2/5*sqrt(6)*sqrt(5), y == 16, l == -1/9*18750^(1/6)]]
> >>
> >> instead of an empty list.
> >
> >When using Maxima (5.45.1) directly I get this result with default settings:
> >
> >(%i1) f: 10*x^(1/3)*y^(2/3)$
> >
> >(%i2) g: 5*x^2 + 6*y$
> >
> >(%i3) solve([diff(f,x)=l*diff(g,x), diff(f,y)=l*diff(g,y), g=120], [x,y,l]);
> >1/6
> >2 sqrt(6) 18750
> >(%o3) [[x = -, y = 16, l = ],
> > sqrt(5) 9
> > 1/6
> >2 sqrt(6)18750
> > [x = - -, y = 16, l = -
> > ]]
> > sqrt(5)9
> >
> >Does Sage modify some Maxima settings related to this or does it call
> >something other than solve?
> >
> >--
> >Oscar
> >
>
> --
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Re: [sage-support] Re: limitations of "solve"?
Oh, I see:
(%i23) domain: complex
;
(%o23) complex
(%i24) f: 10*x^(1/3)*y^(2/3)$
(%i25) g: 5*x^2 + 6*y$
(%i26) solve([diff(f,x)=l*diff(g,x), diff(f,y)=l*diff(g,y), g=120], [x,y,l]);
(%o26)[]
On Sun, 3 Dec 2023 at 14:20, Oscar Benjamin wrote:
>
> What does "set domain to complex" mean in terms of Maxima's settings?
>
> Maxima's solve seems to compute complex solutions by default:
>
> (%i21) solve(x^2 + 1);
> (%o21)[x = - %i, x = %i]
>
> On Sun, 3 Dec 2023 at 13:37, Dima Pasechnik wrote:
> >
> > Yes, Sage modifies the defaults of Maxima, in particular we set domain to
> > complex.
> >
> > On 3 December 2023 12:28:45 GMT, Oscar Benjamin
> > wrote:
> > >On Wed, 29 Nov 2023 at 12:40, Eric Gourgoulhon
> > >wrote:
> > >>
> > >> Le mardi 28 novembre 2023 à 18:25:04 UTC+1, kcrisman a écrit :
> > >>
> > >> Yes. Maxima's attitude is that the square root of negative one is an
> > >> expression which might have multiple values, rather than just picking
> > >> one you hope might be consistent over branch points.
> > >>
> > >> To enforce Maxima to work in the real domain, avoiding to play too much
> > >> with complex square roots, one can add at the beginning of the Sage
> > >> session:
> > >>
> > >> maxima_calculus.eval("domain: real;")
> > >>
> > >> Then the second example in the initial message of this thread yields
> > >>
> > >> [[x == 2/5*sqrt(6)*sqrt(5), y == 16, l == 1/9*18750^(1/6)], [x ==
> > >> -2/5*sqrt(6)*sqrt(5), y == 16, l == -1/9*18750^(1/6)]]
> > >>
> > >> instead of an empty list.
> > >
> > >When using Maxima (5.45.1) directly I get this result with default
> > >settings:
> > >
> > >(%i1) f: 10*x^(1/3)*y^(2/3)$
> > >
> > >(%i2) g: 5*x^2 + 6*y$
> > >
> > >(%i3) solve([diff(f,x)=l*diff(g,x), diff(f,y)=l*diff(g,y), g=120],
> > >[x,y,l]);
> > >1/6
> > >2 sqrt(6) 18750
> > >(%o3) [[x = -, y = 16, l = ],
> > > sqrt(5) 9
> > >
> > > 1/6
> > >2 sqrt(6)18750
> > > [x = - -, y = 16, l = -
> > > ]]
> > > sqrt(5)9
> > >
> > >Does Sage modify some Maxima settings related to this or does it call
> > >something other than solve?
> > >
> > >--
> > >Oscar
> > >
> >
> > --
> > You received this message because you are subscribed to the Google Groups
> > "sage-support" group.
> > To unsubscribe from this group and stop receiving emails from it, send an
> > email to sage-support+unsubscr...@googlegroups.com.
> > To view this discussion on the web visit
> > https://groups.google.com/d/msgid/sage-support/6F4839F2-38B6-40F2-B080-EFCC1C0C3B65%40gmail.com.
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Re: [sage-support] Re: limitations of "solve"?
I did go on to discuss this on the Maxima mailing list:
https://sourceforge.net/p/maxima/mailman/maxima-discuss/thread/CADB8Zm56axVDFXRLbJnxm7xnnbQiixBzg4VX1T91ucj%2B-tuGvA%40mail.gmail.com/#msg58347791
Apparently domain:complex as used by Sage is not a very well tested
configuration of Maxima.
There are most likely other effects besides the documented one about
sqrt(x^2) not simplifying to abs(x):
$ git grep domain src/simp.lisp
src/simp.lisp:(defmvar $limitdomain '$complex)
src/simp.lisp: (cond ((or (and $logexpand (eq $domain '$real))
src/simp.lisp:(cond ((or $numer_pbranch (eq $domain '$complex))
src/simp.lisp: (or (and (eq $domain '$real) (not
(apparently-complex-to-judge-by-$csign-p (cadr gr
src/simp.lisp: (and (eq $domain '$complex)
(apparently-real-to-judge-by-$csign-p (cadr gr)
src/simp.lisp: (or (and (eq $domain '$real) (not
(apparently-complex-to-judge-by-$csign-p (cadr gr
src/simp.lisp: (and (eq $domain '$complex)
(apparently-real-to-judge-by-$csign-p (cadr gr)
src/simp.lisp: (and (eq $domain '$real) $radexpand))
src/simp.lisp: ((or (eq $domain '$complex) (not $radexpand)) (go up)))
src/simp.lisp:(and (eq $domain '$complex)
src/simp.lisp:(and (eq $domain '$real)
src/simp.lisp: (eq $domain '$real))
src/simp.lisp: (eq $domain '$real)
src/simp.lisp: ((and (eq $domain '$real)
src/simp.lisp: (eq $domain '$real)
src/simp.lisp: (and (eq $domain '$real) (ratnump e) (oddp (caddr e)
src/simp.lisp: ((eq $domain '$real)
--
Oscar
On Mon, 1 Jan 2024 at 21:55, Nils Bruin wrote:
>
> The documented effect is usually of most impact:
>
> https://maxima.sourceforge.io/docs/manual/maxima_46.html#index-domain
>
> there may be other undocumented effects, but the one above tends to explain a
> lot already.
>
> On Sunday 3 December 2023 at 06:26:20 UTC-8 Oscar Benjamin wrote:
>>
>> What does "set domain to complex" mean in terms of Maxima's settings?
>>
>> Maxima's solve seems to compute complex solutions by default:
>>
>> (%i21) solve(x^2 + 1);
>> (%o21) [x = - %i, x = %i]
>>
>> On Sun, 3 Dec 2023 at 13:37, Dima Pasechnik wrote:
>> >
>> > Yes, Sage modifies the defaults of Maxima, in particular we set domain to
>> > complex.
>> >
>> > On 3 December 2023 12:28:45 GMT, Oscar Benjamin
>> > wrote:
>> > >On Wed, 29 Nov 2023 at 12:40, Eric Gourgoulhon
>> > >wrote:
>> > >>
>> > >> Le mardi 28 novembre 2023 à 18:25:04 UTC+1, kcrisman a écrit :
>> > >>
>> > >> Yes. Maxima's attitude is that the square root of negative one is an
>> > >> expression which might have multiple values, rather than just picking
>> > >> one you hope might be consistent over branch points.
>> > >>
>> > >> To enforce Maxima to work in the real domain, avoiding to play too much
>> > >> with complex square roots, one can add at the beginning of the Sage
>> > >> session:
>> > >>
>> > >> maxima_calculus.eval("domain: real;")
>> > >>
>> > >> Then the second example in the initial message of this thread yields
>> > >>
>> > >> [[x == 2/5*sqrt(6)*sqrt(5), y == 16, l == 1/9*18750^(1/6)], [x ==
>> > >> -2/5*sqrt(6)*sqrt(5), y == 16, l == -1/9*18750^(1/6)]]
>> > >>
>> > >> instead of an empty list.
>> > >
>> > >When using Maxima (5.45.1) directly I get this result with default
>> > >settings:
>> > >
>> > >(%i1) f: 10*x^(1/3)*y^(2/3)$
>> > >
>> > >(%i2) g: 5*x^2 + 6*y$
>> > >
>> > >(%i3) solve([diff(f,x)=l*diff(g,x), diff(f,y)=l*diff(g,y), g=120],
>> > >[x,y,l]);
>> > > 1/6
>> > > 2 sqrt(6) 18750
>> > >(%o3) [[x = -, y = 16, l = ],
>> > > sqrt(5) 9
>> > > 1/6
>> > > 2 sqrt(6) 18750
>> > > [x = - -, y = 16, l = - ]]
>> > > sqrt(5) 9
>> > >
>> > >Does Sage modify some Maxima settings related to this or does it call
>> > >something other than solve?
>> > >
>> > >--
>> > >Oscar
>> > >
>> >
>> > --
>> > You received this message because you are subscribed to the Google Groups
>> &
Re: [sage-support] Re: Sage seems to incorrectly evaluate fractional powers of complexes
On Thu, 4 Jan 2024 at 12:15, Emmanuel Charpentier
wrote:
>
> But this does not explain whiy Sage is uneble o check (numericalmlmy or
> otherwise) the solutions given by Sympy or Mathematica, which check in Sympy
> (I didn’t yet try to check them in Mathematica, the limitations of the
> current Mathematica interface make this bothersome…).
The problem is with Sage's subs method:
sage: e1 = (-40/3)**(2/3)
sage: e2 = (y**(2/3)).subs({y:-40/3})
sage: e1
4*(-5/3)^(2/3)
sage: e2
4*(5/3)^(2/3)
sage: e1.n()
-2.81144221767250 + 4.86956076355288*I
sage: e2.n()
5.62288443534499
When the number to be substituted has an exact cube factor subs loses
the sign under the radical:
sage: (y^(2/3)).subs({y:-12})
(-12)^(2/3)
sage: (y^(2/3)).subs({y:-24})
4*3^(2/3)
This is most likely due to confusing y^(n/m) as (y^n)^(1/m) when it
should be (y^(1/m))^n. These are not equivalent if y^(1/m) means the
principal mth root.
--
Oscar
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Re: [sage-support] Why cant this limit be evaluated?
On Mon, 19 Feb 2024 at 20:25, Mark “Essa King” Sukaiti < xzark.suk...@gmail.com> wrote: > D(r)=-1/2*((3*r^2 - 1)*elliptic_ec((4*r/(r^2 + 2*r + 1)))^2 + (r^2 - 2*r + > 1)*elliptic_ec((4*r/(r^2 + 2*r + 1)))*elliptic_kc((4*r/(r^2 + 2*r + > 1/(pi^2*r^8 - 2*pi^2*r^7 - pi^2*r^6 + 4*pi^2*r^5 - pi^2*r^4 - > 2*pi^2*r^3 + pi^2*r^2) > > D(r).limit(r=0) > > The limit should be -0.125 (or -1/8) but it seems maxima doesnt know the > limit of elliptic_ec(x) for x->0 > > Sympy also fails giving -5/8. > To follow up on this the SymPy bug is now fixed in master: https://github.com/sympy/sympy/pull/26264 In [1]: e = ((1 - 3*x**2)*elliptic_e(4*x/(x**2 + 2*x + 1))**2/2 - (x**2 - 2*x + 1)*elliptic_e(4*x/(x* ...: *2 + 2*x + 1))*elliptic_k(4*x/(x**2 + 2*x + 1))/2)/(pi**2*x**8 - 2*pi**2*x**7 - pi**2*x**6 + ...: 4*pi**2*x**5 - pi**2*x**4 - 2*pi**2*x**3 + pi**2*x**2) In [2]: e.limit(x, 0) Out[2]: -1/8 Oscar -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/CAHVvXxQst7%3DUUOYd%3DcRAGaMfahH_6LtvV73kwKPvzaC1Dy_Rtw%40mail.gmail.com.
