[sage-devel] Re: Possible bug in gen_legendre_P (associated Legendre polynomials)
Unfortunately, I am not familiar with the details either. Nevertheless, I have made the proposed change for the interval -1https://mathworld.wolfram.com/AssociatedLegendrePolynomial.html. Most importantly, this also solves the problem with spherical harmonics, which was the a highly requested fix. As for resolving the convention conflicts in a rigorous manner I propose a follow-up ticket: https://trac.sagemath.org/ticket/31637. Best, Michael Eric Gourgoulhon schrieb am Sonntag, 8. April 2018 um 18:07:56 UTC+2: > Hi, > > > Le mardi 27 mars 2018 14:46:52 UTC+2, Ralf Stephan a écrit : >> >> >> I think it will suffice for now to put the fact in the documentation. >> > > I am afraid this is not sufficient: a consequence of this bug is that Sage > gives a silly answer for something as elementary as the spherical harmonic > Y_1^1(theta, phi), see > https://trac.sagemath.org/ticket/25034#comment:3 > > Best wishes, > > Eric. > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/aa03efad-b9f4-4cf0-b211-cfc3e5281714n%40googlegroups.com.
[sage-devel] Re: Possible bug in gen_legendre_P (associated Legendre polynomials)
Hi, Le mardi 27 mars 2018 14:46:52 UTC+2, Ralf Stephan a écrit : > > > I think it will suffice for now to put the fact in the documentation. > I am afraid this is not sufficient: a consequence of this bug is that Sage gives a silly answer for something as elementary as the spherical harmonic Y_1^1(theta, phi), see https://trac.sagemath.org/ticket/25034#comment:3 Best wishes, Eric. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
[sage-devel] Re: Possible bug in gen_legendre_P (associated Legendre polynomials)
On Tuesday, March 27, 2018 at 11:20:02 AM UTC+2, James Womack wrote: > > I have created a ticket on Sage trac for this issue: > https://trac.sagemath.org/ticket/25034 > Thanks. > As I mention in the ticket, I think that this issue raises a question as > to whether the Func_assoc_legendre_P class is correctly defined, given that > it now seems to cover both the Ferrers and associated Legendre functions. > Given that the intervals on which the functions are defined do not overlap, > maybe it makes sense to have a single class for both, but to give it amore > generic name? > I think it will suffice for now to put the fact in the documentation. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
[sage-devel] Re: Possible bug in gen_legendre_P (associated Legendre polynomials)
I have created a ticket on Sage trac for this issue:
https://trac.sagemath.org/ticket/25034
As I mention in the ticket, I think that this issue raises a question as to
whether the Func_assoc_legendre_P class is correctly defined, given that it
now seems to cover both the Ferrers and associated Legendre functions.
Given that the intervals on which the functions are defined do not overlap,
maybe it makes sense to have a single class for both, but to give it amore
generic name?
On Thursday, 22 March 2018 20:07:08 UTC, Howard Cohl wrote:
>
> Oh, by the way, Wolfram Mathworld is just completely wrong on this page
> you referenced.
> There is a huge difference between the two functions.
> Also, there is no such thing as an associated Legendre polynomial.
> There is a Legendre polynomials, but if you take the degrees and orders to
> be integers for associated Legendre functions, they don't always end up
> being polynomials, as you can see from the formulas which I showed earlier
> in this thread.
>
> On Thursday, March 22, 2018 at 8:49:16 AM UTC-7, James Womack wrote:
>>
>> Thanks. If that is the case, then presumably this *is* a bug in Sage
>> Math and Func_assoc_legendre_P should distinguish the special cases for
>> n == m when x > 1 or x < 1 when evaluating associated Legendre polynomials.
>>
>> Would you be able to clarify the distinction between Ferrers functions of
>> the first kind and associated Legendre functions for a non-expert? Wolfram
>> Mathworld seems to suggest that they are the same:
>> http://mathworld.wolfram.com/FerrersFunction.html
>>
>> On Thursday, 22 March 2018 15:23:03 UTC, Howard Cohl wrote:
>>>
>>>
>>>
>>> On Thursday, March 22, 2018 at 3:25:06 AM UTC-7, Samuel Lelievre wrote:
Ralf wrote:
> Thanks,
> P.S. Still someone should contact DLMF with the right arguments.
I just emailed them with cc to sage-devel.
>>>
>>> There's nothing wrong with the formula. The Legendre function in the
>>> DLMF is for arguments greater than 1, and is not valid for arguments less
>>> than one. For arguments less than one the correct formula is
>>>
>>> P_m^m(x)=(-1)^m (2m)!/(2^m m!) (1-x^2)^(m/2).
>>>
>>> Both of these are easy to derive using the well-known formulae for
>>> P_\nu^{-\nu} and {\sf P}_\nu^{-\nu} and the connection formulas which
>>> relate P_{\nu}^{-m} to P_{\nu}^m, and for Ferrers functions. See
>>> http://dlmf.nist.gov/14.5.iv and
>>> https://dlmf.nist.gov/14.9.
>>> Where P is the associated Legendre function of the first kind, and {\sf
>>> P} is the Ferrers function of the first kind.
>>>
>>
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[sage-devel] Re: Possible bug in gen_legendre_P (associated Legendre polynomials)
Oh, by the way, Wolfram Mathworld is just completely wrong on this page you
referenced.
There is a huge difference between the two functions.
Also, there is no such thing as an associated Legendre polynomial.
There is a Legendre polynomials, but if you take the degrees and orders to
be integers for associated Legendre functions, they don't always end up
being polynomials, as you can see from the formulas which I showed earlier
in this thread.
On Thursday, March 22, 2018 at 8:49:16 AM UTC-7, James Womack wrote:
>
> Thanks. If that is the case, then presumably this *is* a bug in Sage Math
> and Func_assoc_legendre_P should distinguish the special cases for n == m
> when x > 1 or x < 1 when evaluating associated Legendre polynomials.
>
> Would you be able to clarify the distinction between Ferrers functions of
> the first kind and associated Legendre functions for a non-expert? Wolfram
> Mathworld seems to suggest that they are the same:
> http://mathworld.wolfram.com/FerrersFunction.html
>
> On Thursday, 22 March 2018 15:23:03 UTC, Howard Cohl wrote:
>>
>>
>>
>> On Thursday, March 22, 2018 at 3:25:06 AM UTC-7, Samuel Lelievre wrote:
>>>
>>> Ralf wrote:
>>> > Thanks,
>>> > P.S. Still someone should contact DLMF with the right arguments.
>>>
>>> I just emailed them with cc to sage-devel.
>>>
>>
>> There's nothing wrong with the formula. The Legendre function in the DLMF
>> is for arguments greater than 1, and is not valid for arguments less than
>> one. For arguments less than one the correct formula is
>>
>> P_m^m(x)=(-1)^m (2m)!/(2^m m!) (1-x^2)^(m/2).
>>
>> Both of these are easy to derive using the well-known formulae for
>> P_\nu^{-\nu} and {\sf P}_\nu^{-\nu} and the connection formulas which
>> relate P_{\nu}^{-m} to P_{\nu}^m, and for Ferrers functions. See
>> http://dlmf.nist.gov/14.5.iv and
>> https://dlmf.nist.gov/14.9.
>> Where P is the associated Legendre function of the first kind, and {\sf
>> P} is the Ferrers function of the first kind.
>>
>
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[sage-devel] Re: Possible bug in gen_legendre_P (associated Legendre polynomials)
The Ferrers functions are defined on the real segment (-1,1).
The associated Legendre functions are in general defined on the Complex
plane except for the ray (-\infty,1].
Typically Ferrers functions are written with argument x=\cos\theta, |x|<1
and associated Legendre functions are written with argument z=\cosh\eta,
z>1, or something to that effect. One can obtain the Ferrers functions
through a limiting process from the associated Legendre functions by taking
an appropriately weighted average of x+i0, and x-i0, ,where x\in(-1,1).
If you read the DLMF chapter on associated Legendre functions, then all
this is explained quite clearly. The DLMF uses different symbols for the
associated Legendre functions vs. the Ferrers functions. An italic P and Q
for the associated Legendre functions and a sans serif P and Q for the
Ferrers functions. Their definitions and properties are all given in that
DLMF chapter. In Abramowitz and Stegun, the same symbol is used for both
functions, but you can tell they are using which by the argument z (for
associated Legendre) vs. x (for Ferrers functions).
If you have any other questions, please free to email me hcohl -at- nist
dot gov.
On Thursday, March 22, 2018 at 8:49:16 AM UTC-7, James Womack wrote:
>
> Thanks. If that is the case, then presumably this *is* a bug in Sage Math
> and Func_assoc_legendre_P should distinguish the special cases for n == m
> when x > 1 or x < 1 when evaluating associated Legendre polynomials.
>
>
Would you be able to clarify the distinction between Ferrers functions of
> the first kind and associated Legendre functions for a non-expert? Wolfram
> Mathworld seems to suggest that they are the same:
> http://mathworld.wolfram.com/FerrersFunction.html
>
> On Thursday, 22 March 2018 15:23:03 UTC, Howard Cohl wrote:
>>
>>
>>
>> On Thursday, March 22, 2018 at 3:25:06 AM UTC-7, Samuel Lelievre wrote:
>>>
>>> Ralf wrote:
>>> > Thanks,
>>> > P.S. Still someone should contact DLMF with the right arguments.
>>>
>>> I just emailed them with cc to sage-devel.
>>>
>>
>> There's nothing wrong with the formula. The Legendre function in the DLMF
>> is for arguments greater than 1, and is not valid for arguments less than
>> one. For arguments less than one the correct formula is
>>
>> P_m^m(x)=(-1)^m (2m)!/(2^m m!) (1-x^2)^(m/2).
>>
>> Both of these are easy to derive using the well-known formulae for
>> P_\nu^{-\nu} and {\sf P}_\nu^{-\nu} and the connection formulas which
>> relate P_{\nu}^{-m} to P_{\nu}^m, and for Ferrers functions. See
>> http://dlmf.nist.gov/14.5.iv and
>> https://dlmf.nist.gov/14.9.
>> Where P is the associated Legendre function of the first kind, and {\sf
>> P} is the Ferrers function of the first kind.
>>
>
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[sage-devel] Re: Possible bug in gen_legendre_P (associated Legendre polynomials)
Thanks. If that is the case, then presumably this *is* a bug in Sage Math
and Func_assoc_legendre_P should distinguish the special cases for n == m
when x > 1 or x < 1 when evaluating associated Legendre polynomials.
Would you be able to clarify the distinction between Ferrers functions of
the first kind and associated Legendre functions for a non-expert? Wolfram
Mathworld seems to suggest that they are the same:
http://mathworld.wolfram.com/FerrersFunction.html
On Thursday, 22 March 2018 15:23:03 UTC, Howard Cohl wrote:
>
>
>
> On Thursday, March 22, 2018 at 3:25:06 AM UTC-7, Samuel Lelievre wrote:
>>
>> Ralf wrote:
>> > Thanks,
>> > P.S. Still someone should contact DLMF with the right arguments.
>>
>> I just emailed them with cc to sage-devel.
>>
>
> There's nothing wrong with the formula. The Legendre function in the DLMF
> is for arguments greater than 1, and is not valid for arguments less than
> one. For arguments less than one the correct formula is
>
> P_m^m(x)=(-1)^m (2m)!/(2^m m!) (1-x^2)^(m/2).
>
> Both of these are easy to derive using the well-known formulae for
> P_\nu^{-\nu} and {\sf P}_\nu^{-\nu} and the connection formulas which
> relate P_{\nu}^{-m} to P_{\nu}^m, and for Ferrers functions. See
> http://dlmf.nist.gov/14.5.iv and
> https://dlmf.nist.gov/14.9.
> Where P is the associated Legendre function of the first kind, and {\sf P}
> is the Ferrers function of the first kind.
>
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[sage-devel] Re: Possible bug in gen_legendre_P (associated Legendre polynomials)
On Thursday, March 22, 2018 at 3:25:06 AM UTC-7, Samuel Lelievre wrote:
>
> Ralf wrote:
> > Thanks,
> > P.S. Still someone should contact DLMF with the right arguments.
>
> I just emailed them with cc to sage-devel.
>
There's nothing wrong with the formula. The Legendre function in the DLMF
is for arguments greater than 1, and is not valid for arguments less than
one. For arguments less than one the correct formula is
P_m^m(x)=(-1)^m (2m)!/(2^m m!) (1-x^2)^(m/2).
Both of these are easy to derive using the well-known formulae for
P_\nu^{-\nu} and {\sf P}_\nu^{-\nu} and the connection formulas which
relate P_{\nu}^{-m} to P_{\nu}^m, and for Ferrers functions. See
http://dlmf.nist.gov/14.5.iv and
https://dlmf.nist.gov/14.9.
Where P is the associated Legendre function of the first kind, and {\sf P}
is the Ferrers function of the first kind.
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[sage-devel] Re: Possible bug in gen_legendre_P (associated Legendre polynomials)
Thanks. I am waiting for an account on Sage trac, then I will submit a bug report. On Thursday, 22 March 2018 10:25:06 UTC, Samuel Lelievre wrote: > > Ralf wrote: > > Thanks, > > P.S. Still someone should contact DLMF with the right arguments. > > I just emailed them with cc to sage-devel. > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
[sage-devel] Re: Possible bug in gen_legendre_P (associated Legendre polynomials)
Ralf wrote: > Thanks, > P.S. Still someone should contact DLMF with the right arguments. I just emailed them with cc to sage-devel. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
[sage-devel] Re: Possible bug in gen_legendre_P (associated Legendre polynomials)
arb agrees here: sage: CBF(1/2).legendre_P(1,1) [-0.8660254037844386 +/- 5.90e-17] So I'd suggest using complex balls for your numerics until the bug is fixed. Thanks, P.S. Still someone should contact DLMF with the right arguments. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
