symbolic Legendre / associated Legendre functions / polynomials

[rws]

Defect because there is no Sage binding for a result returned by Maxima:

sage: hypergeometric([-1/2,1/2],[2],4).simplify_hypergeometric()
1/2*I*sqrt(3)*assoc_legendre_p(1/2, -1, -5/3)
sage: assoc_legendre_p
...
NameError: name 'assoc_legendre_p' is not defined

Existing numeric functions are legendre_P, legendre_Q, gen_legendre_P, and gen_legendre_Q which correspond to P(n,x) / Q(n,x) and associated Legendre P(n,m,x) / Q(n,m,x).

They should be made symbolic. FLINT has fast code for P(n,x).

• PQ(n,x)
  • ​https://en.wikipedia.org/wiki/Legendre_polynomials
  • ​http://mathworld.wolfram.com/LegendrePolynomial.html
• PQ(n,m,x)
  • ​https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
  • ​http://mathworld.wolfram.com/AssociatedLegendrePolynomial.html
• functions PQ
  • ​https://en.wikipedia.org/wiki/Legendre_function
  • ​http://mathworld.wolfram.com/LegendreFunctionoftheFirstKind.html
  • ​http://mathworld.wolfram.com/LegendreFunctionoftheSecondKind.html

See also ​http://ask.sagemath.org/question/27230/problem-with-hypergeometric/

[maldun]

Hi!

Good to see that someone else is working on making the orthogonal polynomials symbolic, since my research interests shifted heavily in the past years.

A good read on Legendre polynomials is also the bible for ortho polys: Abramowitz and Stegun ​http://people.math.sfu.ca/~cbm/aands/page_331.htm

There you will find also much information on special values and other properties. I currently have some time left and can help with one thing or the other if you like,

[rws]

Fine! See also ​http://trac.sagemath.org/wiki/symbolics/functions

[rws]

OK I have a question. What is the equivalent recursive algorithm to ​https://github.com/sagemath/sage/blob/master/src/sage/functions/orthogonal_polys.py#L812-834 for Legendre polynomials?

The link is valid as long #16812 is not merged.

[maldun]

Replying to rws:

OK I have a question. What is the equivalent recursive algorithm to ​https://github.com/sagemath/sage/blob/master/src/sage/functions/orthogonal_polys.py#L812-834 for Legendre polynomials?

The link is valid as long #16812 is not merged.

Hi!

You won't have luck to find an equivalent recursion algorithm for Legendre Polynomials, since the recursion algorithm for Chebyshev Polynomials uses the fact that cheby polynomials are cosines in disguise, and thus one is able to build Cheby polyis in O(log N) time. For Legendre polynomials you have to use the classic recursion formula given in ​https://en.wikipedia.org/wiki/Legendre_polynomials#Recursive_definition

[maldun]

Maybe this could help you:

I already implemented all Orthopolys one time: ​http://trac.sagemath.org/attachment/ticket/9706/trac_9706_ortho_polys.patch

they only would need cleanup/restructuring. Maybe you can reuse some of the implemented methods (like recursions and derivatives)

[rws]

Some timings for P(n,z):

sage: legendre_P(100,2.5)
6.39483750487443e66
sage: timeit('legendre_P(100,2.5)')
25 loops, best of 3: 21 ms per loop

sage: from mpmath import legenp 
sage: legenp(100,0,2.5)
mpf('6.3948375048744286e+66')
sage: timeit('legenp(100,0,2.5)')
625 loops, best of 3: 97.2 µs per loop

sage: from scipy.special import eval_legendre
sage: eval_legendre(int(100),float(2.5))
6.3948375048744324e+66
sage: timeit('eval_legendre(int(100),float(2.5))')
625 loops, best of 3: 7.62 µs per loop

sage: eval_legendre(int(10^5),float(1.00001))
3.1548483029540554e+192
sage: timeit('eval_legendre(int(10^6),float(2.5))')
25 loops, best of 3: 11.8 ms per loop
sage: eval_legendre(int(10^6),float(2.5))
inf

while legenp will already bail out at 10⁵ because of F convergence issues.

[rws]

With P(n,x) symbolics and algebra, Pari is much better than Maxima

sage: P.<t> = QQ[]
sage: timeit('legendre_P(1000,t)')
5 loops, best of 3: 2.8 s per loop
sage: timeit('pari.pollegendre(1000,t)')
625 loops, best of 3: 366 µs per loop

[rws]

This is a proof of concept patch, and one can already use legendre_P and see from that and the code how the other three functions will look like. So, now would be a good time for fundamental criticism 8)

---

New commits:

​50da8c5	16813: skeleton P(n,x)
​e74539b	16813: P(n,x) refined, documentation
​0f86b77	16813: fixes for doctest failures

[rws]

Replying to maldun:

A good read on Legendre polynomials is also the bible for ortho polys: Abramowitz and Stegun ​http://people.math.sfu.ca/~cbm/aands/page_331.htm

This appears outdated, it is replaced by ​http://dlmf.nist.gov/14

[maldun]

Replying to rws:

Replying to maldun:

A good read on Legendre polynomials is also the bible for ortho polys: Abramowitz and Stegun ​http://people.math.sfu.ca/~cbm/aands/page_331.htm

This appears outdated, it is replaced by ​http://dlmf.nist.gov/14

You can't call a source outdated, which still covers information that the newer source doesn't. I checked your link, and some things from A&S are missign e.g. explicit representation of Legendre Polynomials with their polynomial coefficients. And on another note: I don't see much harm in citing a classic work on this topic ...

[rws]

Replying to maldun:

I already implemented all Orthopolys one time: ​http://trac.sagemath.org/attachment/ticket/9706/trac_9706_ortho_polys.patch

they only would need cleanup/restructuring. Maybe you can reuse some of the implemented methods (like recursions and derivatives)

I am not sure about the derivatives. For P(3,2,x).diff(x) I get -45*x^2 + 15 (Wolfram agrees) while with your formula (lines 2377-2395 of the patch) I get (after simplification) -45*x^2 - 15.

Update: what's your reference there?

[maldun]

Replying to rws:

Replying to maldun:

I already implemented all Orthopolys one time: ​http://trac.sagemath.org/attachment/ticket/9706/trac_9706_ortho_polys.patch

they only would need cleanup/restructuring. Maybe you can reuse some of the implemented methods (like recursions and derivatives)

I am not sure about the derivatives. For P(3,2,x).diff(x) I get -45*x^2 + 15 (Wolfram agrees) while with your formula (lines 2377-2395 of the patch) I get (after simplification) -45*x^2 - 15.

Update: what's your reference there?

It seems you are right. from Gradshteyn-Ryzhik p.1004 formula 8.731-1 we have the relation

P(n,m,x).diff(x) = ((n+1-m)*P(n+1,m,x)-(n+1)*x*P(n,m,x))/(x**2-1)

The same relation holds for gen_legendre_Q

I suppose that's an copy/paste/rewrite mistake from my side.

[rws]

Also, your recursive functions for Q(n,x) and Q(n,m,x) appear to be wrong:

sage: legendre_Q.eval_recursive(2,x).subs(x=3)
13/2*I*pi + 13/2*log(2) - 9/2
sage: legendre_Q.eval_recursive(2,x).subs(x=3).n()
0.00545667363964419 + 20.4203522483337*I
sage: legendre_Q(2,3.)
0.00545667363964451 - 20.4203522483337*I

The latter result from mpmath is supported by Wolfram.

As to Q(n,m,x):

sage: gen_legendre_Q(2,1,x).subs(x=3)
-1/8*sqrt(-2)*(72*I*pi + 72*log(2) - 50)
sage: gen_legendre_Q(2,1,x).subs(x=3).n()
39.9859464434253 + 0.0165114736149170*I
sage: gen_legendre_Q(2,1,3.)
-39.9859464434253 + 0.0165114736149193*I

Again, Wolfram supports the latter value from mpmath (symbolic as (25 i)/(2 sqrt(2))-18 i sqrt(2) ((log(4))/2+1/2 (-log(2)-i pi))).

[rws]

OK, I resolved it by using conjugate() on every logarithm in the Q(n,x) algorithms (on which the Q(n,m,x) recurrence is based, too).

Update: it however makes symbolic work tedious and differentiation impossible, at the moment.

See also ​https://groups.google.com/forum/?hl=en#!topic/sage-support/bEMPMEYeZKU on derivatives of conjugates in Sage.

[maldun]

Thanks for resolving this issue! I suppose I wasn't careful enough with complex arguments. But in my defense: I hadn't time to test this codes well enough when I wrote them ... but hopefully they give some useful informations.

concerning complex conjugation: I hope my answer on the mailing list give some clues: ​https://groups.google.com/forum/?hl=en#!topic/sage-support/bEMPMEYeZKU

Replying to rws:

OK, I resolved it by using conjugate() on every logarithm in the Q(n,x) algorithms (on which the Q(n,m,x) recurrence is based, too).

Update: it however makes symbolic work tedious and differentiation impossible, at the moment.

See also ​https://groups.google.com/forum/?hl=en#!topic/sage-support/bEMPMEYeZKU on derivatives of conjugates in Sage.

[rws]

Replying to maldun:

Thanks for resolving this issue!

Unfortunately, while it would be easy to resolve this numerically, the instances of conjugate() introduced in the recurrence will make symbolic results from the recurrence unreadable and, in case of derivatives, impossible to use. A different way of computing the recurrences is needed, one which does away with usage of conjugate().

[maldun]

Hi!

We have several possible ways out of this: 1) avoid recursion for symbolic argument for Legendre_Q and use another library (maxima, flint, sympy ... ) for evaluation. 2) Let it be, but avoid it as default method. 3) Maybe more elegantly: There is a more closed relation for legendre_Q (but it's not really a recursion):

Q(n,z) = ½P(n,z) ln((z+1)/(z-1)) - W(n-1,z)

with

W(n,z) = Σ_{k=1}^n (1/k) P(k-1,z) P(n-k,z)

(Gradshteyn-Ryzhik p 1019f)

Maybe we could find an recursion for W(n,z)

Hope this could be of some use

Edit there should be a recursion since W(n,z) is the convolution of aseries of holonomic functions, and if I remeber correctly there is an theorem saying that convolutions of holonomic functions are also holonomic, thus should have a recursion.

Edit: The above mentioned relation can also be found here: ​http://people.math.sfu.ca/~cbm/aands/page_334.htm

[rws]

sage: from ore_algebra import *
sage: def W(n):
    return sum([(1/k)*legendre_P(k-1,t)*legendre_P(n-k,t) for k in range(1,n+1)])
....: 
sage: R.<t> = QQ['t']
sage: guess([W(n) for n in range(1,10)], OreAlgebra(R['n'], 'Sn'))
(-n - 3)*Sn^2 + (2*t*n + 5*t)*Sn - n - 2

[maldun]

Nice! I didn't know that sage already supports Ore Algebras. It appears that my holonomic function package on Mathematica stopped to work.

[rws]

I always need some time to figure out the final form (that offset of n!) but: n*W_n = (2tn-t)*W_n-1 - (n-1)W_n-2 (W₀=0, W₁=1).

[rws]

However, this will yield the same result unless the log has conjugate associated with it. This shows your recurrence is actually correct but numerical results derived from it by substitution may need a warning about the log branch. I know not enough about calculus, maybe I'll ask again, this time on sage-devel, if there should be a switch for log() to select the branch in case of numerical evaluation. What do you think?

[rws]

I think there must be another different formula, because Wolfram has this for Q(2,x):

sage: ((3*x^2-1)/2*(log(x+1)-log(1-x))/2-3*x/2).subs(x=3)
-13/2*I*pi + 13/2*log(4) - 13/2*log(2) - 9/2
sage: ((3*x^2-1)/2*(log(x+1)-log(1-x))/2-3*x/2).subs(x=3).n()
0.00545667363964419 - 20.4203522483337*I

which yields the correct value without use of conjugate.

The first few Q(n,x) from Wolfram are:

Q(0,x) = 1/2 log(x+1)-1/2 log(1-x)
Q(1,x) = x (1/2 log(x+1)-1/2 log(1-x))-1
Q(2,x) = 1/2 (3 x^2-1) (1/2 log(x+1)-1/2 log(1-x))-(3 x)/2
Q(3,x) = -(5 x^2)/2-1/2 (3-5 x^2) x (1/2 log(x+1)-1/2 log(1-x))+2/3

which makes clear that instead of log((1+x)/(1-x)).conjugate() we should just use log(1+x)-log(1-x), of course. Oh well.

[maldun]

Oh yeah it's again the non uniqueness of the representation of the complex logarithm

sage: log((x+1)/(1-x)).subs(x=3)
I*pi + log(2)
sage: (log(x+1)-log(1-x)).subs(x=3).simplify_log()
-I*pi + log(2)
sage: log((x+1)/(1-x)).subs(x=3).conjugate()
-I*pi + log(2)

confusing as hell ...

I think Wolfram uses the log(1+x)-log(1-x) representation simply by the fact that it is independent of the branch in the following sense: Let log(x) = ln|x| + i*arg(x) + 2kπi and log(y) = ln|y| + i*arg(y) + 2kπi then

log(x) - log(y) = ln|x| + i*arg(x) + 2kπi- ln|y| + i*arg(y) + 2kπi = 
= ln|x/y| + i*(arg(x) - arg(y)) + 0 

I.e. if we have the same branch on the logarithm the module of 2kπi cancels out.

That means the formula isn't exactly wrong, it uses simply a different branch of the logarithm. But the representation of log as difference saves us indeed a lot of trouble, and as showed above is independent of the branch we use.

Nevertheless I think we should stick to the recursion with W(n,x), because from a computational view it is a lot better since:

1) The computational complexity is the same (solving a two term recursion)

2) we save computation time since we don't have to simplify expressions containing logarithms but only polynomials which are much simpler to handle and expand.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​0b67667	16813: Q(n,x) implementation
​06df7c8	16813: implement P(n,m,x)
​7bd4a70	Merge branch 'develop' into t/16813/symbolic_legendre___associated_legendre_functions___polynomials
​74ca8ea	16813: implement Q(n,m,x); fixes, doctests

[rws]

Replying to maldun:

Nevertheless I think we should stick to the recursion with W(n,x), because from a computational view it is a lot better since:

1) The computational complexity is the same (solving a two term recursion)

2) we save computation time since we don't have to simplify expressions containing logarithms but only polynomials which are much simpler to handle and expand.

Well, I have implemented your recurrence using multivariate polynomials where the generator l stands for the log term and gets substituted subsequently. This is already twice as fast as Maxima. However, your intuition was right that the W(n,x) formula is still faster, my guess because univariate polys are faster than multi. Note that the P(n,x) have to be computed, too, but nevertheless it's about 10x the speed of Maxima (which BTW uses the wrong log branch as well).

I might add some introductory doc cleanup but the functions themselves are now finished. Please review.

[maldun]

Replying to rws:

Well, I have implemented your recurrence using multivariate polynomials where the generator l stands for the log term and gets substituted subsequently. This is already twice as fast as Maxima. However, your intuition was right that the W(n,x) formula is still faster, my guess because univariate polys are faster than multi. Note that the P(n,x) have to be computed, too, but nevertheless it's about 10x the speed of Maxima (which BTW uses the wrong log branch as well).

I might add some introductory doc cleanup but the functions themselves are now finished. Please review.

If Maxima uses also this branch of the logarithm we should make sure that changing the branch of the logarithm does not interfere with existing code. Have you already testet the complete sage library with

sage -testall

?

We should also ask on the mailing list if there are some objections with that.

Personally I'm fine with both, as long as it is consistent, since using another branch of the logarithm is not wrong, but maybe not expected. (Maybe I programmed the recursion that way, since I compared it with Maxima that time, so that the output does not change)

[rws]

Replying to maldun:

Have you already testet the complete sage library with

sage -testall

?

Buildbot task is queued.

[maldun]

Replying to rws:

Replying to maldun:

Have you already testet the complete sage library with

sage -testall

?

Buildbot task is queued.

I asked on sage-devel if there are other objections on using the log(1+z) - log(1-z) representation: ​https://groups.google.com/forum/?hl=en#!topic/sage-devel/5_4Pr8GypUA Let's see what the other developers think.

[rws]

I'm afk for some time, will see when I get back.

[rws]

Replying to maldun:

Replying to rws:

Replying to maldun:

Have you already testet the complete sage library with

sage -testall

?

Buildbot task is queued.

And tested successfully, see the top of the ticket.

[maldun]

Very good! Since there are no objections on sage-devel this ticket only needs review now.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​93915cc	Merge branch 'develop' into t/16813/symbolic_legendre___associated_legendre_functions___polynomials
​daf47c5	16813: adaptations due to new BuiltinFunction code

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​779ea3a	Merge branch 'develop' into t/16813/symbolic_legendre___associated_legendre_functions___polynomials
​a3c7c1e	16813: cosmetics

[mmezzarobba]

I'm getting several doctest failures like:

File "src/sage/functions/orthogonal_polys.py", line 1426, in sage.functions.orthogonal_polys.Func_legendre_Q._maxima_init_evaled_
Failed example:
    legendre_Q._maxima_init_evaled_(20,x).coeff(x^10)
Expected:
    -29113619535/131072*log(-(x + 1)/(x - 1))
Got:
    doctest:1: DeprecationWarning: coeff is deprecated. Please use coefficient instead.
    See http://trac.sagemath.org/17438 for details.
    -29113619535/131072*log(-(x + 1)/(x - 1))

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​a062018	Merge branch 'develop' into t/16813/symbolic_legendre___associated_legendre_functions___polynomials
​4e99dfa	16813: fix doctests

[mmezzarobba]

Sorry if that's a stupid question, but why do you need to override __call__? In any case I guess a comment explaining the reason would be useful.

[mmezzarobba]

sage: legendre_P(0, 1).n()
...
AttributeError: 'int' object has no attribute 'n'
sage: legendre_P(1, 1).n()
1.00000000000000

[mmezzarobba]

sage: legendre_P(42, 12345678)
1464081544112412716366892468459853695115358209756840823628776385121841706762162962766108364297738809627122469598911070029409071998031560780937314580877929546448067606814039101080853578172130265741673137107647826759931295833598386722228898959000304822089300102623241891719774710215869248045233381461475507593850687226480251/549755813888
sage: legendre_P(42, RR(12345678))
+infinity
sage: legendre_P(42, Reals(100)(12345678))
2.6631488146675341638308827323e309

→ Consistent so far. But:

sage: legendre_P(42, Reals(20)(12345678))
legendre_P(42, 1.2346e7)

[mmezzarobba]

Perhaps related to the previous comment, I'm no fan of the mechanism used to choose real_parent. Do you have an example where this code would be useful that could not be handled at the level of Function or perhaps OrthogonalPolynomial?

[mmezzarobba]

More on the wishlist side of things: The numerical evaluation methods return nonsense in cases where Maple, say, is accurate.

sage: legendre_P(201/2, 0).n()
365146.687569733
sage: legendre_P(201/2, 0).n(100).n()
0.0561386178630179

[mmezzarobba]

sage: legendre_P(x, x, x)
...
TypeError: Symbolic function legendre_P takes exactly 2 arguments (3 given)

but:

sage: legendre_P(1, x, x)
x

[mmezzarobba]

Why does Func_legendre_P.__call__ contain:

       elif algorithm == 'recursive':
           return self.eval_recursive(n, x)

while Func_legendre_P has no method eval_recursive?

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​47f5fa0	16813: remove __call__ methods; improve doctests
​07814ad	Merge branch 'develop' into t/16813/symbolic_legendre___associated_legendre_functions___polynomials
​2cc59c6	16813: remove real_parent stuff
​9df78de	16813: more doctests

[rws]

This addresses all issues. Indeed the changes were necessary, thanks for your input.

If you set one of the ortho poly tickets to positive, then please wait with the other, so I can remove merge conflicts.

[mmezzarobba]

Thank you for fixing all this.

One more remark: the main module docstring needs updating (at least to remove the TODO about associated Legendre polynomials, perhaps also to clarify what relies on Maxima and what doesn't).

[mmezzarobba]

Also, like in the case of #17151, I think having:

sage: legendre_P(0, x).parent()
Integer Ring
sage: legendre_P(0, SR(1)).parent()
Integer Ring

is a bug.

Compare for example:

sage: sin(0).parent() # not sure I like this, but somehow makes sense
Integer Ring
sage: sin(SR(0)).parent()
Symbolic Ring

[mmezzarobba]

legendre_P does not know that P(n, 1) == 1.

[mmezzarobba]

I find it strange that Func_legendre_Q and Func_assoc_legendre_{P,Q}derive from OrthogonalPolynomial since these functions are not polynomials, even for fixed integer m and n. And in fact the same argument could apply to Func_legendre_P itself, since they seem to work for non-integer n, even though only the case of integer n appears to be documented.

[mmezzarobba]

Not sure what to think about that:

sage: legendre_P(10, polygen(CC, 'x'))
46189/256*x^10 - 109395/256*x^8 + 45045/128*x^6 - 15015/128*x^4 + 3465/256*x^2 - 63/256
sage: legendre_P(10, polygen(CC, 'x'), algorithm='pari')
180.425781250000*x^10 - 427.324218750000*x^8 + 351.914062500000*x^6 - 117.304687500000*x^4 + 13.5351562500000*x^2 - 0.246093750000000
sage: legendre_P(10, polygen(CC, 'x'), coerce=False)
...
TypeError: arguments must be symbolic expressions
sage: legendre_P(10, polygen(CC, 'x'), algorithm='pari', coerce=False)
180.425781250000*x^10 - 427.324218750000*x^8 + 351.914062500000*x^6 - 117.304687500000*x^4 + 13.5351562500000*x^2 - 0.246093750000000
sage: legendre_P(10, polygen(RIF, 'x'))
46189/256*(x)^10 - 109395/256*(x)^8 + 45045/128*(x)^6 - 15015/128*(x)^4 + 3465/256*(x)^2 - 63/256

Is the idea that calls to legendre_P(n, x) with no algorithm keyword will always take the branch corresponding to x in SR, but the user can obtain faster evaluations at integers etc. by specifying algorithm='pari' is they know what they are doing?

[rws]

Replying to mmezzarobba:

I find it strange that Func_assoc_legendre_{P,Q}derive from OrthogonalPolynomial since these functions are not polynomials, even for fixed integer m and n.

​https://en.wikipedia.org/wiki/Associated_Legendre_polynomials "...they satisfy the orthogonality condition..."

[rws]

Of course, I'd rather have it all under functions/holonomic/...

[mmezzarobba]

Replying to rws:

Replying to mmezzarobba:

I find it strange that Func_assoc_legendre_{P,Q}derive from OrthogonalPolynomial since these functions are not polynomials, even for fixed integer m and n.

​https://en.wikipedia.org/wiki/Associated_Legendre_polynomials "...they satisfy the orthogonality condition..."

Sure, but they are not polynomials. It is necessarily wrong to use the class OrthogonalPolynomial for things that are not, strictly speaking, orthogonal polynomials if really necessary, but then there should at least be a clear warning in the documentation of that class.

Replying to rws:

Of course, I'd rather have it all under functions/holonomic/...

I'm not sure being holonomic or not makes a big difference for a symbolic function. What do you have in mind?

(As for me, I would like to have an implementation of general holonomic functions in Sage at some point, but I'm more thinking of a parent entirely separate from SR and with conversion methods to and from symbolic functions. Symbolic functions that happen to be holonomic could have a method that returns their representation using a system of linear functional equations, and could use that to implement operations for which no better function-specific code is available.)

[mmezzarobba]

sage: legendre_Q(-1., 0.)
(+infinity)*sqrt(pi)*sin(0.500000000000000*pi)

[mmezzarobba]

sage: legendre_Q(1, 1, algorithm='pari')
...
AttributeError: 'Func_legendre_Q' object has no attribute 'eval_pari'

but

sage: legendre_Q(-1, 1, algorithm=pari)
...
TypeError: __call__() got an unexpected keyword argument 'algorithm'

[mmezzarobba]

sage: legendre_Q(-1, x)
...
UnboundLocalError: local variable 'help3' referenced before assignment

[mmezzarobba]

I find it surprising that:

sage: legendre_Q(0, x)
1/2*log(x + 1) - 1/2*log(-x + 1)
sage: legendre_Q(0, pi)
1/2*log(pi + 1) - 1/2*log(-pi + 1)

but:

sage: legendre_Q(0, 2)
legendre_Q(0, 2)

and even:

sage: legendre_Q(0, SR(2))
legendre_Q(0, 2)

[mmezzarobba]

A fun one:

sage: a = legendre_Q(0, 1/2)
sage: a.n()
0.549306144334055
sage: maple(a).evalf()
.5493061443-1.570796327*I

I think the issue is that Maple uses non-standard branch cuts for the Legendre functions, so that the conversion to Maple is not correct.

[rws]

Replying to mmezzarobba:

Is the idea that calls to legendre_P(n, x) with no algorithm keyword will always take the branch corresponding to x in SR, but the user can obtain faster evaluations at integers etc. by specifying algorithm='pari' is they know what they are doing?

Effectively, one of the __call__ methods (at this point of time OrthogonalPolynomial.__call__) dispatches to the resp. eval_... method. If no match is found (and algorithm is None) __call__ returns to somewhere in the BuiltinFunction/Function classes to either return a held object or call evalf. Jeroen wanted to implement a default list of algorithm/eval_... pairs checked automatically but, if this keeps coming up, I'll put it high on my list.

I'm not sure being holonomic or not makes a big difference for a symbolic function. What do you have in mind?

It's just a natural category for special functions, corresponding to a D-finite series expansion.

[rws]

Replying to mmezzarobba:

Also, like in the case of #17151, I think having:

sage: legendre_P(0, x).parent()
Integer Ring
sage: legendre_P(0, SR(1)).parent()
Integer Ring

is a bug.

This is now #17953.

[rws]

From the ask.sagemath issue there is also this discrepancy:

sage: -1/2*sqrt(3)*gen_legendre_P(1/2, -1, -5/3)
-0.483843755630126 + 0.369716687246133*I

vs. Maple:
A := z -> exp(I*Pi*z)*hypergeom([-z,1/2],[2],4):
evalf(A(1/2)); -0.3697166872 + 0.4838437556 I

[rws]

Replying to mmezzarobba:

Is the idea that calls to legendre_P(n, x) with no algorithm keyword will always take the branch corresponding to x in SR, but the user can obtain faster evaluations at integers etc. by specifying algorithm='pari' is they know what they are doing?

No, there is no idea, I just haven't figured out why the BuiltinFunction mechanism won't give me the original x, so I klugded something in OrthogonalFunction.__call__ triggered by algorithm. This will have to be fixed but not today.

[rws]

@mezzarobba: The problem with non-numeric arguments not getting through unconverted to function::eval cannot be solved with the current BuiltinFunction. Some numerics aren't even converted but throw an error, see #17790. I thus think that the given examples with polygen(CC) as argument cannot be implemented at the moment without kludging a taylor-made __call__ method, and should be worked around using substitution. A solution will only be possible with #18832.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​e437940	Merge branch 'develop' into t/16813/symbolic_legendre___associated_legendre_functions___polynomials
​c3e0525	16813: brought introduction up to date
​96e5e01	16813: rename base classes
​32cc796	16813: no more algorithm kwd; small fixes

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​21fc2aa

16813: more cases with Q(n,x)

[chapoton]

needs rebase, does not apply

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​8748825

Merge branch 'develop' into t/16813/symbolic_legendre___associated_legendre_functions___polynomials

[rws]

I have absolutely no idea why there is a merge conflict in orthogonal_polys.py when no one did a change there in the develop branch. Will see if this continues.

[rws]

Squashed it all into one commit.

---

New commits:

​f511634

16813: symbolic Legendre / associated Legendre functions / polynomials

[chapoton]

some failing doctests, see bot report

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​fdf55be	Merge branch 'u/rws/16813-1' of git://trac.sagemath.org/sage into tmp08
​b70ad8d	16813: fix doctests

[rws]

The failure in symbolic/expression_conversions.py is due to a random bug and we now depend on #19464.

[rws]

Replying to rws:

@mezzarobba: The problem with non-numeric arguments not getting through unconverted to function::eval cannot be solved with the current BuiltinFunction. Some numerics aren't even converted but throw an error, see #17790. I thus think that the given examples with polygen(CC) as argument cannot be implemented at the moment without kludging a taylor-made __call__ method, and should be worked around using substitution. A solution will only be possible with #18832.

I have opened #20312 on which the above depends.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​60b8649

Merge branch 'develop' into t/16813/16813-1

[rws]

Maybe it's better to factor out the associated function stuff, regardless if Python or C++.

[mmezzarobba]

Time to get this merged, I guess. I'm still not 100% happy with how evaluation behaves (ideally, in the cases where the function is a polynomial, one would like to be able to evaluate it on elements of any ℚ-algebra), but getting it right in all cases looks difficult, and imperfect code is more useful than no code at all :-).

[rws]

Agree and thanks for the review.

[tscrim]

Needs rebase.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​cfccbb7	Merge branch 'develop' into t/16813/16813-1
​7238198	fix doctest

[rws]

Review needed for the last two commits.

[tscrim]

One failing doctest, see patchbot.

[rws]

Does not fail here. Note the patchbot is running 7.5beta1 not 2, and there were gegenbauer changes with pynac-0.7.0. I'll try to trigger the bots.

[tscrim]

I ended up accidentally trying this on my beta1 as well. All is good.

[vbraun]

The dependency is invalid/wontfix. What are you depending on?

[rws]

Replying to vbraun:

The dependency is invalid/wontfix. What are you depending on?

Actually I don't recall and couldn't find out how floor(x,hold=True) was fixed. That bug was the reason for the random_expr() doctest fail in this and some other ticket. I'll remove the dependency.