Opened 10 years ago
Last modified 15 months ago
#13720 needs_work enhancement
Scale legendre_P to [a,b]
Reported by:  mjo  Owned by:  burcin 

Priority:  major  Milestone:  sageduplicate/invalid/wontfix 
Component:  symbolics  Keywords:  
Cc:  Merged in:  
Authors:  Reviewers:  
Report Upstream:  N/A  Work issues:  
Branch:  Commit:  
Dependencies:  Stopgaps: 
Description (last modified by )
The Legendre polynomials, returned by legendre_P()
, of the first kind are orthogonal over [1,1] and are normalized to have value +1 at the endpoints.
When solving leastsquares problems, it's convenient to be able to construct them over an arbitrary interval [a,b].
Attachments (3)
Change History (49)
comment:1 Changed 10 years ago by
 Status changed from new to needs_review
comment:2 followup: ↓ 3 Changed 10 years ago by
 Status changed from needs_review to needs_work
comment:3 in reply to: ↑ 2 Changed 10 years ago by
Replying to fwclarke:
 The code seems overcomplicated to me. Replacing the last few lines with
R = PolynomialRing(ZZ, 't') f = R([(1)^(n  k)*binomial(n, k)*binomial(n + k, k) for k in range(n + 1)]) return f((x  a)/(b  a))is much simpler and is significantly faster. Moreover, this makes expressions such aslegendre_P(4, Zmod(5), 3, 6)evaluate correctly; at it stands the patch yields an
ArithmeticError: 0^0 is undefined.
I tried this, but it's producing the wrong results. For example,
sage: legendre_P(1, x) 1/2*x  5/2
That said, there are two reasons it might seem overcomplicated.
The first is that I was very careful not to break any existing doctests, even though they're doing some crazy things. I tried to comment each of these hurdles.
The second is that I wanted to be clear about what was happening. The c(m)
and g(m)
functions are as defined in A&S. So the final return
statement is exactly the form given in the reference. The use of the affine transform phi()
allows me to retain that form, and also provides some rationale for the result. I think this makes it (more) clear that we're composing the standard P()
over [1,1] with an affine transform, which will intuitively preserve orthogonality.
 I don't see why
a
andb
need to be real numbers. Actually they could be polynomial variables, giving:sage: R.<r,s,t> = QQ[] sage: legendre_P(2, t, r, s) 6*((r  t)/(r  s)  1)*(r  t)/(r  s) + 1
I'm happy to change this if it can be done without breaking the existing doctests. I'll wait to see what you say about the first thing.
 I don't see the need to use
bool
in doctests such asbool(legendre_P(0, x, 0, 1) == p0)
Often, the ==
operator will return a symbolic equality rather than True/False?. This is true even in this simple case:
sage: p0 = 1 sage: legendre_P(0, x, 0, 1) == p0 1 == 1
If you cast the symbolic equality to bool
, you get,
 True, if Sage can convince itself that equality holds.
 False, if Sage can "prove" inequality.
 False, if it's not sure.
So it should be safe to test for True
.
 It's not clear to me why backquotes are used in some of the error strings:
raise TypeError('`n` must be a natural number')but
raise ValueError('n must be nonnegative')
I was just careless with that one, I'll fix it. I think backticks or single quotes are needed around 'a' and 'b', otherwise it reads weird. But 'n' at the beginning is fine without it.
comment:4 followup: ↓ 5 Changed 10 years ago by
 Status changed from needs_work to needs_review
I've fixed the variable quoting in the new patch, and allowed the endpoints a,b to be symbolic. There's also a new test to make sure that the symbolic endpoints work as expected.
I haven't heard back, so maybe you bought my reasoning for the affine transform =)
The bool()
casts do have to stay, otherwise the tests just don't work.
Thanks for taking a look.
comment:5 in reply to: ↑ 4 ; followup: ↓ 6 Changed 9 years ago by
 Status changed from needs_review to needs_work
Replying to mjo:
I've fixed the variable quoting in the new patch, and allowed the endpoints a,b to be symbolic. There's also a new test to make sure that the symbolic endpoints work as expected.
Why force them to be symbolic? Better, surely, to allow anything that will evaluate and let coercion handle it.
I haven't heard back, so maybe you bought my reasoning for the affine transform =)
No, just busy with other things. There is an affine transform in what I wrote, but it's a transform of the interval [a, b] onto [0, 1] rather than [1, 1]. This is simpler, and the Legendre polynomials on the interval [0, 1] have a simple form.
I tried this, but it's producing the wrong results. For example,
sage: legendre_P(1, x) 1/2*x  5/2
This is what results from using ^
rather than **
in python. I'm sorry my proposed code had only been testing in Sage, and therefore was being prepared. This dealt with, my version does pass all the doctests, is faster when if the variable is a polynomial generator, and the error
sage: legendre_P(4, Zmod(5)(1), 3, 6) ... ArithmeticError: 0^0 is undefined.
is eliminated. Apologies for mistyping this before.
The
bool()
casts do have to stay, otherwise the tests just don't work.
Another way of doing this is to introduce a polynomial variable. Thus
sage: R.<x> = QQ[] sage: legendre_P(1, x) == x True
I know very little about the symbolic side of Sage, but I do find things like
sage: SR(1) == SR(1) 1 == 1
very strange, and confusing when compared with
sage: 1 == 1 True
comment:6 in reply to: ↑ 5 Changed 9 years ago by
Replying to fwclarke:
Why force them to be symbolic? Better, surely, to allow anything that will evaluate and let coercion handle it.
I removed the RR check, so you can actually pass in anything you want now. I'm not convinced this isn't a footgun, but it's still a clear improvement over what we have now so I'm happy with it.
This is what results from using
^
rather than**
in python.
Ah, yes, I should have caught that, sorry.
I'm sorry my proposed code had only been testing in Sage, and therefore was being prepared. This dealt with, my version does pass all the doctests, is faster when if the variable is a polynomial generator, and the error
sage: legendre_P(4, Zmod(5)(1), 3, 6) ... ArithmeticError: 0^0 is undefined.is eliminated. Apologies for mistyping this before.
I believe this is another bug, elsewhere. I've set out to fix it and am trapped in a rabbit hole at the moment, about four bugs from here. Once that's sorted out, I'll do some experiments.
The
bool()
casts do have to stay, otherwise the tests just don't work.Another way of doing this is to introduce a polynomial variable. Thus
sage: R.<x> = QQ[] sage: legendre_P(1, x) == x TrueI know very little about the symbolic side of Sage, but I do find things like
sage: SR(1) == SR(1) 1 == 1very strange, and confusing when compared with
sage: 1 == 1 True
You'll get no argument from me there. I do prefer to test with the default symbol x
though just because that's what most people will use.
comment:7 followup: ↓ 8 Changed 9 years ago by
I think that people on this ticket may find #9706 interesting, though I imagine it has bitrotted somewhat and Burcin had a very long list of comments for it.
comment:8 in reply to: ↑ 7 Changed 9 years ago by
 Status changed from needs_work to needs_review
Replying to kcrisman:
I think that people on this ticket may find #9706 interesting, though I imagine it has bitrotted somewhat and Burcin had a very long list of comments for it.
That's too big to ever get merged... I would break out the functions into individual tickets where possible.
Anyway, I was waiting here on #13786 which fixes the ArithmeticError: 0^0 is undefined
mentioned above. Unfortunately, both implementations now fail due to some other error on legendre_P(4, Zmod(5)(1), 3, 6)
, but what can you do.
Now the only difference between my implementation and the one in comment 2 is speed: mine is slightly faster unscaled, and that one is slightly faster on a nonstandard interval. But both deltas are negligible, and I have an aesthetic preference for the algorithm that's described in the reference.
This still passes all the tests and is a big improvement over what we have so I'm going to put it up for review again.
comment:9 Changed 9 years ago by
Brief questions:
 When specialcasing
n=0
, you havereturn 1
Will this be a Python int?  What if
n
is a Python int?We can accept Python integers for ``x`` as well::
 How's speed? I can imagine the sum thing getting slow.
 Am I misreading this? Should it be beta = 1?
# From Abramowitz & Stegun, (22.3.2) with alpha = beta = 0.
Oh, I see. It's actually (22.3.1) and maybe this should be in the code with a reference section.
comment:10 Changed 9 years ago by
 Reviewers set to Francis Clarke, KarlDieter Crisman
Changed 9 years ago by
comment:11 Changed 9 years ago by
When n=0, I try to return unity from the same ring as the variable:
if n == 0: # Easy base case, save time. Attempt to return a value in the # same field/ring as x. try: return x.parent()(1) except AttributeError: # In case something without a parent was given for x. return 1
The exceptional case is just there as a failsafe, and it would probably give a python int. I guess ZZ(1) is better, so I've changed it.
If n
is a python int, we're happy. Early in the function we check that n
can be coerced into ZZ
. Then later,
# Ensure that (2**n) is an element of ZZ. This is used later  # we can divide finite field elements by integers but we can't # multiply them by rationals at the moment. n = ZZ(n)
For speed, the current implementation has,
sage: timeit('legendre_P(100,x)') 5 loops, best of 3: 57.1 ms per loop
The posted patch has,
sage: timeit('legendre_P(100,x)') 5 loops, best of 3: 47.3 ms per loop
So, even for the unscaled version it's an improvement. There's a slowdown for the scaled version, of course:
sage: timeit('legendre_P(100,x,5,10)') 5 loops, best of 3: 70.4 ms per loop
But it's still very fast.
You're right about the reference, I must have typo'd it. It's fixed, and I've added a REFERENCES section to the docs.
The new patch is up, I forgot to check the box, so use the one with the "2" at the end.
comment:12 Changed 9 years ago by
 Status changed from needs_review to needs_work
Thanks. I'm glad I asked, because before
sage: legendre_P(0,float(1)) 1 sage: type(_) <type 'int'>
but now
sage: legendre_P(0,float(1)) 1 sage: type(_) <type 'sage.rings.integer.Integer'>
Work:
 So to my surprise, plotting these things works great. It would be nice to add one or two examples.
 I wouldn't ask for this on this ticket, but it probably should have one or two examples of the errors actually catching silly input. I find it worthwhile to test those branches.
Is that okay? Otherwise I think this is nice.
comment:13 Changed 9 years ago by
Oh, I should ask the dumb question of whether this scaling is standard in terminology, or whether we need any other disclaimers.
comment:14 Changed 9 years ago by
 Description modified (diff)
 Status changed from needs_work to needs_review
Ok, I added a plotting example, and three tests for nonsense input. They're at the end of their respective sections.
What might not be standard terminology? "Scaling"? In any case I don't think I'm qualified to say, but it made sense to me at the time.
comment:15 Changed 9 years ago by
Quick update to use tmp_filename() per Jeroen's comment on #12852.
comment:16 Changed 9 years ago by
 Milestone changed from sage5.11 to sage5.12
comment:17 Changed 9 years ago by
apply only sagetrac_13720.patch
Changed 9 years ago by
comment:18 followup: ↓ 19 Changed 9 years ago by
 Status changed from needs_review to needs_work
I really don't understand why a
and b
get converted into symbolic expressions. This has some very strange consequences, e.g.,
sage: legendre_P(3, 7, 1/2, 1/2).parent() Symbolic Ring
It is my understanding that coercion should arrange that the parent of an expression is as close as possible to the parents of the constituant parts. It is thus wrong to force (almost) everything into the Symbolic Ring.
But anyway, I'm afraid I still prefer my version of the code: (1) for its simplicity (having 3 local functions, two of them only used once, seems far too overelaborate); (2) my code is significantly faster.
If you don't like that then something like
if a == 1 and b == 1: _init() return sage_eval(maxima.eval('legendre_p(%s,x)'%ZZ(n)), locals={'x':x}) else: return legendre_P(n, (2*x  a  b)/(b  a), 1, 1)
would be a very simple change to the existing code which avoids 'reinventing the wheel'. This would also have the advantage that an almost identical change would provide scaled versions for the the Legendre functions legendre_Q
of the second kind. However the maxima code is very slow.
Incidentally, in all cases there needs to be a check to see if a == b
.
comment:19 in reply to: ↑ 18 Changed 8 years ago by
Replying to fwclarke:
I really don't understand why
a
andb
get converted into symbolic expressions. This has some very strange consequences, e.g.,sage: legendre_P(3, 7, 1/2, 1/2).parent() Symbolic RingIt is my understanding that coercion should arrange that the parent of an expression is as close as possible to the parents of the constituant parts. It is thus wrong to force (almost) everything into the Symbolic Ring.
I had a comment in the code about this, but I've forgotten the details. In any case  whatever the issue was  it seems to be fixed. I removed the manual coercions of n,a,b and all of the tests still pass.
So this is fixed:
sage: legendre_P(3, 7, 1/2, 1/2).parent() Rational Field
Incidentally, in all cases there needs to be a check to see if
a == b
.
This will throw a dividebyzero. What's the alternative?
comment:20 Changed 8 years ago by
This will throw a dividebyzero. What's the alternative?
Nevermind, stupid question =)
comment:21 Changed 8 years ago by
 Branch set to u/mjo/ticket/13720
 Commit set to ce0a4578af9efdef9356d318d698daed1ef27e9b
 Description modified (diff)
 Status changed from needs_work to needs_review
comment:23 Changed 8 years ago by
 Milestone changed from sage6.1 to sage6.2
comment:24 Changed 8 years ago by
 Status changed from needs_review to needs_work
This needs to be rebased.
comment:25 Changed 8 years ago by
 Commit changed from ce0a4578af9efdef9356d318d698daed1ef27e9b to 36b4c29920bc4cfbc09b8b7a1fd9de757f8179d3
Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:
36b4c29  Trac #13720: Allow scaling of legendre_P to an arbitrary interval [a,b].

comment:26 Changed 8 years ago by
 Status changed from needs_work to needs_review
I had to change the _init()
doctest to use the laguerre polynomials instead, since the implementation of legendre_P()
no longer uses maxima.
comment:27 followup: ↓ 28 Changed 8 years ago by
I'm not sure if they're related, but the Maxima
assoc_legendre_p[v,u] (z) Legendre function of degree v and order u assoc_legendre_q[v,u] (z) Legendre function, 2nd kind
from the symbolic wiki on Trac could be relevant here. If so, we should have conversions  if not, sorry for the noise.
comment:28 in reply to: ↑ 27 ; followup: ↓ 29 Changed 8 years ago by
Replying to kcrisman:
I'm not sure if they're related, ...
Those two functions are different than our legendre_{P,Q}
. Maxima has matching legendre_p(n,x)
and legendre_q(n,x)
which we're currently using for our implementations.
I updated the note on the wiki page.
comment:29 in reply to: ↑ 28 Changed 8 years ago by
I'm not sure if they're related, ...
Those two functions are different than our
legendre_{P,Q}
. Maxima has matchinglegendre_p(n,x)
andlegendre_q(n,x)
which we're currently using for our implementations.I updated the note on the wiki page.
Thanks, and I updated based on that update, very helpful.
comment:30 Changed 8 years ago by
 Milestone changed from sage6.2 to sage6.3
comment:31 Changed 8 years ago by
 Status changed from needs_review to needs_work
patchbot:
sage t long src/sage/functions/orthogonal_polys.py # 33 doctests failed
comment:32 Changed 8 years ago by
 Branch changed from u/mjo/ticket/13720 to u/chapoton/13720
 Commit changed from 36b4c29920bc4cfbc09b8b7a1fd9de757f8179d3 to 1de3cf47ac8d5d696d2ccf8f1e46281eba4564b4
 Status changed from needs_work to needs_review
comment:33 Changed 8 years ago by
 Milestone changed from sage6.3 to sage6.4
comment:34 Changed 7 years ago by
 Commit changed from 1de3cf47ac8d5d696d2ccf8f1e46281eba4564b4 to 0843342b41d1ce501cab9b84c89e2ef1d84e5dba
Branch pushed to git repo; I updated commit sha1. New commits:
0843342  Merge branch 'u/chapoton/13720' into 6.5.b2

comment:35 Changed 7 years ago by
Thanks for cleaning that up, I had forgotten about it.
comment:36 Changed 7 years ago by
Conflicts with #16813. We might want to merge efforts.
comment:37 Changed 7 years ago by
 Milestone changed from sage6.4 to sage6.8
comment:38 Changed 7 years ago by
 Branch changed from u/chapoton/13720 to u/kdilks/13720
comment:39 Changed 7 years ago by
 Branch changed from u/kdilks/13720 to u/chapoton/13720
 Milestone changed from sage6.8 to sage6.10
comment:40 Changed 7 years ago by
 Branch changed from u/chapoton/13720 to u/kdilks/13720
 Commit changed from 0843342b41d1ce501cab9b84c89e2ef1d84e5dba to 8d1354cf63bcca9b01733e4f57aaa16556e3ba0f
New commits:
8d1354c  fixed merge conflict

comment:41 Changed 7 years ago by
 Status changed from needs_review to needs_work
You introduced merge conflict markers in your branch. Better always eyeball your diff before committing.
comment:42 Changed 7 years ago by
Yeesh, yeah, thought it was just the one thing, but a lot more needs to be cleaned up. I'll take care of that tomorrow.
comment:43 Changed 7 years ago by
 Branch changed from u/kdilks/13720 to u/chapoton/13720
 Commit changed from 8d1354cf63bcca9b01733e4f57aaa16556e3ba0f to 0843342b41d1ce501cab9b84c89e2ef1d84e5dba
I'll just change it back and let Frédéric deal with rebasing.
comment:44 Changed 15 months ago by
 Milestone changed from sage6.10 to sageduplicate/invalid/wontfix
 Reviewers changed from Francis Clarke, KarlDieter Crisman to Michael Orlitzky
 Status changed from needs_work to positive_review
Let's just kill this. I'm not going to rewrite the whole function from scratch, as would be necessary at this point.
comment:45 followup: ↓ 46 Changed 15 months ago by
 Branch u/chapoton/13720 deleted
 Commit 0843342b41d1ce501cab9b84c89e2ef1d84e5dba deleted
 Reviewers Michael Orlitzky deleted
 Status changed from positive_review to needs_work
Can't someone else start from scratch, though? It seems like a worthwhile ticket to have open.
comment:46 in reply to: ↑ 45 Changed 15 months ago by
Replying to kcrisman:
Can't someone else start from scratch, though? It seems like a worthwhile ticket to have open.
It would be tricky since legendre_P
is now implemented as a BuiltinFunction
. You would need a different symbolic function for each interval [a,b]
because, for example, its symbolic derivative changes when the interval does. But it's probably doable somehow.
I was the original reporter and also the author. If I remember correctly, I needed it to simplify some finite element example problem, and these days I have no need for it. But naturally I wouldn't object to someone else making an attempt.
a
andb
need to be real numbers. Actually they could be polynomial variables, giving:bool
in doctests such as