Make Airy functions symbolic

[olazo]

As discussed in ​sage-support.

Currently sage can evaluate airy functions numerically:

sage: airy_ai(1.4)
0.0820380498076

but it doesn't work symbolically

sage: airy_ai(x)
...
TypeError: Cannot evaluate symbolic expression to a numeric value.

We should make it symbolical for both airy_ai and airy_bi, as well as their derivatives.

[olazo]

I've added a patch, which should do the job, but it has a few shortcomings:

1.-The resulting symbolic functions seem to remain on hold:

sage: airy_ai(1.0)
airy_ai(1.00000000000000)

You need to force it to evaluate:

sage: airy_ai(1.0).n()
0.135292416313

2.- This doesn't work:

sage: airy_ai(2.0).n(digits=100)
0.0349241304233

3.- There is no evaluation for airy_ai_prime or airy_bi_prime

4.- I'm not sure about how should the functions be called, some possible schemes are

{ai,bi,aip,bip} {ai,bai,aip,baip} {airy_ai,airy_bi,airy_ai_prime,airy_bi_prime}

And also whether the latex representation should be capitalized or not. I chose the third scheme, and capitalized typesetting.

[kcrisman]

Good start. A few responses.

• You'll need to have more methods. We should probably use ​mpmath to evaluate these functions in general, though using ​the SciPy version could be useful over RDF. See #11888 for a very recent example of this kind of thing and how to implement it.
• That should also fix precision issues, if properly done, I think.
• SciPy has evaluation for the airy primes. I think that Maxima has these too. (Cc:ing FJ in case he has plans to add these to mpmath.)
• We should have translations of these to Maxima and friends. That will probably help us name them. What do Mathematica and Maple call them?
• All methods, including initialization methods, need doctests. Many of the new symbolic functions added at wiki:symbolics/functions will have nice models to follow.

Thanks for getting us a start on this! That's great.

[fredrik.johansson]

mpmath has the derivatives (as well as integrals) -- just use the optional 'derivative' parameter.

[kcrisman]

mpmath has the derivatives (as well as integrals) -- just use the optional 'derivative' parameter.

Ok, I didn't realize that was what that parameter was about. Though in retrospect it seems obvious!

[olazo]

On second thought, I think it would be better to use the airy equation to calculate derivatives or order higher than 1. Like

sage: airy_ai(2,x)
x*airy_ai(x)
sage: airy_ai(3,x)
airy_ai(x)+x*airy_ai_prime(x)
sage: diff(airy_ai(x),x,2)
x*airy_ai(x)
sage: diff(airy_ai(x),x,3)
airy_ai(x)+x*airy_ai_prime(x)

which is very likey to be the way mpmath calculates higher order derivatives. Integrals however, would be returned as:

sage: airy_ai(-1,x)
airy_ai(-1,x)
sage: integral(airy_ai(x),x)
airy_ai(-1,x)

what do you think?

[olazo]

I just added a new patch. The new version includes generalized derivatives, evaluation with mpmath, and special values of the functions and their derivatives. Just a one doubt, the coverage is:

oscar@oscar-netbook:~$ sage -coverage airy.py
SCORE /home/oscar/sage/my_patches/airy.py: 76% (20 of 26)

Missing documentation:
	 * __init__(self):
	 * __init__(self):
	 * __init__(self):
	 * __init__(self):
	 * __init__(self):
	 * __init__(self):


Possibly wrong (function name doesn't occur in doctests):
	 * _derivative_(self, x, diff_param=None):
	 * _eval_(self, x):
	 * _evalf_(self, x, parent=None):
	 * _derivative_(self, x, diff_param=None):
	 * _eval_(self, x):
	 * _evalf_(self, x, parent=None):
	 * _derivative_(self, alpha, *args, **kwds):
	 * _eval_(self, alpha, *args):
	 * _evalf_(self, alpha, x, parent=None):
	 * _derivative_(self, x, diff_param=None):
	 * _eval_(self, x):
	 * _evalf_(self, x, parent=None):
	 * _derivative_(self, x, diff_param=None):
	 * _eval_(self, x):
	 * _evalf_(self, x, parent=None):
	 * _derivative_(self, alpha, *args, **kwds):
	 * _eval_(self, alpha, *args):
	 * _evalf_(self, alpha, x, parent=None):

Is that important?

[kcrisman]

Is that important?

The second part isn't, as you are implicitly testing these 'hidden' functions. By the way,

Examples::

should be

EXAMPLES::

However, the first part (the initialization methods) is important for our coverage. You can just do a couple from the big examples that you have.

Do these pass doctests? I'm surprised that

sage: airy_ai_simple= FunctionAiryAiSimple()

would work when you run tests, assuming you didn't import FunctionAiryAiSimple into the global namespace.

Anyway, overall on a quick glance this looks great; unfortunately, I won't have time to review it properly soon - hopefully someone else will, because you put a lot of great work into it and we should totally have these symbolic. Thanks!

[olazo]

Do these pass doctests? I'm surprised that

sage: airy_ai_simple= FunctionAiryAiSimple()

would work when you run tests, assuming you didn't import FunctionAiryAiSimple into the global namespace.

It looks like they do:

oscar@oscar-netbook:~$ sage -coverage airy.py
sage -t  "/home/oscar/sage/my_patches/airy.py"              
	 [51.5 s]
 
----------------------------------------------------------------------
All tests passed!
Total time for all tests: 51.7 seconds

Why shouldn't they? I'll take care of the init functions.

[olazo]

I'm sorry, that should be:

oscar@oscar-netbook:~$ sage -t airy.py
sage -t  "/home/oscar/sage/my_patches/airy.py"              
	 [51.5 s]
 
----------------------------------------------------------------------
All tests passed!
Total time for all tests: 51.7 seconds

(the command was sage -t)

[kcrisman]

sage: airy_ai_simple= FunctionAiryAiSimple()

would work when you run tests, assuming you didn't import FunctionAiryAiSimple into the global namespace.

I'm just surprised this doesn't cause trouble.

On another note, this apparently does something weird when applied to 5.0.beta4. Namely, Sage won't start:

---> 65 from sage.functions.all import sin, cos

ImportError: cannot import name sin
Error importing ipy_profile_sage - perhaps you should run %upgrade?
WARNING: Loading of ipy_profile_sage failed.

I'll spare you the rest of the traceback. I think there is some kind of circular import thing going on here, but I don't understand imports that well. Perhaps moving the import of airy to further down (after trig functions) would help - that's a totally uninformed guess, of course.

[olazo]

I just added a new patch, now with 100 % test coverage, but I still get this message:

oscar@oscar-netbook:~$ sage -coverage /home/oscar/sage/my_patches/airy.py ----------------------------------------------------------------------
/home/oscar/sage/my_patches/airy.py
ERROR: Please add a `TestSuite(s).run()` doctest.
SCORE /home/oscar/sage/my_patches/airy.py: 100% (26 of 26)
----------------------------------------------------------------------

What is this TestSuite? thing? I also changed the order in which functions are initialized, so that airy_ai_general is initialized first. Doing it last made some symbolic operations, such as simplify not work (It said something about airy_ai requiring 2 arguments).

[benjaminfjones]

Hi Oscar, I haven't looked over your patch very closely yet, but I plan to. One comment on first glance: in several places you call

return mpmath_utils.call(airy_bi_mpmath, x, derivative=1, parent=RR) 

but I think we want to preserve the parent of x instead of forcing it to be RR. Importing the parent function from sage.structure.coerce using a different name like sage_parent (because as burcin pointed out in one of my tickets, parent is the name of the parameter in _evalf_) and doing something like

from sage.structure.coerce import parent as sage_parent
R = parent or sage_parent(x)
return mpmath_utils.call(airy_bi_mpmath, x, derivative=1, parent=R)

would do the trick I think.

Other comment: can we add conversions to Maxima along with the Mma conversion you included? Here is a link to the appropriate chapter in the manual: ​http://maxima.sourceforge.net/docs/manual/en/maxima_15.html#SEC80

[benjaminfjones]

The patch applies to 5.0.beta6 cleanly, but upon sage -br I get an import error ending with:

/home/jonesbe/sage/sage-5.0.beta6/local/lib/python2.7/site-packages/sage/gsl/dft.py in <module>()
     63 from sage.rings.real_mpfr import RR
     64 from sage.rings.all import I
---> 65 from sage.functions.all import sin, cos
     66 from sage.gsl.fft import FastFourierTransform
     67 from sage.gsl.dwt import WaveletTransform

ImportError: cannot import name sin
Error importing ipy_profile_sage - perhaps you should run %upgrade?
WARNING: Loading of ipy_profile_sage failed.

Seems like the same thing kcrisman ran into with the previous patch applied to 5.0.beta4.

[benjaminfjones]

I found the cause of the problem in comment 17. There is a circular import going on because

from airy import airy_ai, airy_bi

occurs at the top of sage/functions/all.py. If you move it to the bottom of the file, sage starts just fine.

There a several doctests that fail, however. Oscar, I think that running sage -coverage only checks to see if there are doctest for every function, it doesn't actually run the tests. Use

sage -t devel/sage/sage/functions/airy.py

to run the tests and get feedback.

[benjaminfjones]

Here at SD 40.5, we're going to pick this ticket up, do the work, and get it into sage.

[dsm]

Okay, I'm posting this for future reference. I've attempted to preserve as much as possible of the original code while switching to the "new-style" dictionary-argument .n() approach we're working on in #12289, but I'm setting it to "needs work" myself while we think through some of the consequences of the switch. Some of the _evalf_ stuff violates DRY and I'm not yet sure of the best way to deal with it.

As well, the patch currently segfaults in testing when it computes beta(0.5, 0.5) due to some problem with the current #12289 package, which I suspect is unrelated. But I think the new approach is beginning to take shape.

[dsm]

newstyle rewrite, version 1

[dsm]

Oh, yeah: and #13028 was a significant nuisance while working on this.

[eviatarbach]

New patch, which should pass all doctests once #12289 is merged.

It makes the following changes:

• Changing an incorrect airy_bi identity
• Adding airy.py to the documentation
• Checking whether n is an integer (for some reason mpmath is able to evaluate numerically with non-integral n)
• Formatting changes
• Removing the parent's precision check because it wasn't used
• Removing the check for the parent keyword because Burcin told me to

[kcrisman]

Needs review, then? Note that I had to rebase #12289.

[kcrisman]

Patchbot apply trac_12455-newstyle-airy-rebase.patch and trac_12455_newstyle_airy2.patch

I don't know that either of the needs work issues are still there...

[kcrisman]

This needs a trivial extra patch to fix

sage -t sage/functions/special.py
**********************************************************************
File "sage/functions/special.py", line 389, in sage.functions.special._init
Failed example:
    spherical_harmonic(3,2,1,2)
Expected:
    15/4*sqrt(7/30)*e^(4*I)*sin(1)^2*cos(1)/sqrt(pi)
Got:
    15/4*sqrt(7/30)*cos(1)*e^(4*I)*sin(1)^2/sqrt(pi)
**********************************************************************

which is really a rebase issue off of #9880, no worries.

[kcrisman]

Okay, a little more rebasing for #12289. Patches coming up.

[kcrisman]

Apply trac_12455-newstyle-airy-rebase.patch and trac_12455-newstyle-airy2-rebase.patch

[kcrisman]

Eviatar, how much would you say still needs to be reviewed? (In the sense that you have not yet given it positive review.) Just your patch?

(I'm running doctests now.)

[kcrisman]

I made some mistake while rebasing - one moment.

[eviatarbach]

I think I was fairly thorough, but I haven't positive-reviewed since my patch hasn't been looked at by someone else.

[kcrisman]

Okay, now I have to fix something in the other one... sigh. Final version coming soon, it does pass tests!

[kcrisman]

Patchbot, apply trac_12455-newstyle-airy-rebase.patch and trac_12455-newstyle-airy2-rebase.patch

[kcrisman]

(for some reason mpmath is able to evaluate numerically with non-integral n)

But this is even in the documentation you wrote!

Return the `\alpha`-th order fractional derivative with respect to `z`.

"Fractional", right? So ... ? Plus, you do not make this change in airy_bi, only airy_ai, for some reason.

Another question:

sage: (plot(airy_bi(x), (x, -10, 5)) +\ 

Are we allowing this kind of continuation in doctests still? I can't remember but seem to recall something about this being a problem.

Otherwise this all seems fine.

[eviatarbach]

I didn't write it. But looking at the mpmath documentation, it should work. It didn't work for most values before my patch anyway, but I'll see what I can do.

[kcrisman]

I didn't write it.

Good point, sorry! In fact, it's just copied from ​the mpmath doc. But now you have taken charge :-) Unfortunately, I can't find a lot of independent implementations of this...

[burcin]

trac_12455-airy_review.patch​ fixes a couple of problems:

• airy_{a,b}i_general should evaluate to airy_{a,b}i_simple, airy_{a,b}i_prime, etc. depending on the first parameter

• differentiating airy_{a,b}i_general in the first parameter is not allowed

• _evalf_ should not raise errors for unrecognized algorithm argument

This is still needs work because we use indirect doctests everywhere. There is no reason to go through the airy_{a,b}i wrapper functions. We should call the symbolic functions directly in the doctests. There are also a lot of code paths that are not tested. I changed the fractional order behavior, but didn't need to change any doctests.

It would be great if someone else can fix/add doctests. I will move on to something else now.

[eviatarbach]

This patch depends on a new release of Pynac and other fixes to make the algorithm keyword work sensibly. Also, I don't see much point in having the Maxima implementation as an option, considering the fact that it doesn't accept complex input, doesn't have arbitrary precision, and is slower than mpmath (doesn't appear to scale better either).

I'd hate to delay this patch longer, especially considering that the Airy functions are probably among the most-used special functions. Could we remove the Maxima option for now and perhaps add it in the future, maybe along with a patch that adds a SciPy? option as suggested in one of the comments?

[benjaminfjones]

I'm fine with removing Maxima from the options for numerical evaluation (numerical is not generally where Maxima shines). I agree with @burcin, though, that indirect doctests should be fixed wherever possible and become direct tests and that the branch coverage should be higher.

What exactly is the issue with having the Maxima option? Is it the limited precision and domain? We could always check that inputs are okay whenever Maxima is selected and raise an exception otherwise.

[git]

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

​aeb0800	12455: remove algorithm=maxima
​22731c9	12455: implement algorithm='scipy'

[rws]

Replying to benjaminfjones:

What exactly is the issue with having the Maxima option? Is it the limited precision and domain? We could always check that inputs are okay whenever Maxima is selected and raise an exception otherwise.

It's useless, especially when scipy is implemented as algorithm. OTOH, it introduces more code paths to test and deepens a dependency that is unwanted.

Oh, and scipy is a bit faster:

sage: timeit("airy_bi(2).n(algorithm='maxima')")
625 loops, best of 3: 685 µs per loop
sage: timeit("airy_bi(2).n(algorithm='scipy')")
625 loops, best of 3: 124 µs per loop

[git]

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

​a88d035	12455: fix indirect and plot doctests
​f91a3c2	12455: hold_derivate only implemented in wrapper functions
​515e343	12455: add small index to documentation head

[rws]

Everything was reviewed up to my recent changes:

• merged develop
• removal of maxima and inclusion of scipy for numerics
• replacement of indirect with direct doctests (still 100% coverage)
• removal of hold_derivate keyword from airy_ai/bi_general() where it wasn't implemented (it is still in airy_ai/bi() wrapper functions
• small index at the top of documentation for orientation, because of confusing presentation order of functions, which cannot be controlled, sadly

So please review.

[chapoton]

one failing doctest, see patchbot report

[git]

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

​1715892	Merge branch 'develop' into t/12455/make_airy_functions_symbolic
​8e76828	12455: trigger doctest with MaximaFunction as intended

[chapoton]

Rebased on 6.4.beta3, and add a few # rel tol to the scipy results.

---

New commits:

​c3449e8	Merge branch 'u/rws/make_airy_functions_symbolic' and 6.4.beta3
​9552606	trac #12455 fixing doctests by adding rel tol and changing imports

[git]

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

​7102227

Merge with 6.4.beta4

[git]

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

​5b731d3

trac #12455 change use of Airy as example of maxima function

[git]

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

​39deab4

trac #12455 formatting doc and code

[rws]

sage -t --long src/sage/functions/airy.py
**********************************************************************
File "src/sage/functions/airy.py", line 404, in sage.functions.airy.airy_ai
Failed example:
    plot(airy_ai(x), (x, -10, 5)) + plot(airy_ai_prime(x),
     (x, -10, 5), color='red')
Expected nothing
Got:
    Graphics object consisting of 2 graphics primitives
**********************************************************************
File "src/sage/functions/airy.py", line 799, in sage.functions.airy.airy_bi
Failed example:
    plot(airy_bi(x), (x, -10, 5)) + plot(airy_bi_prime(x),
     (x, -10, 5), color='red')
Expected nothing
Got:
    Graphics object consisting of 2 graphics primitives

[git]

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

​fbd6c35

trac #12455 plot output

[jdemeyer]

Just to make you aware: this conflicts with #17130. When either of these tickets gets positive_review, the other should be rebased.

[jdemeyer]

And why the change to def hypergeometric_U? The new code seems quite hackish, why is it needed and how does it relate to this ticket?

[jdemeyer]

These should be an error:

sage: airy_ai(3).n(algorithm='scipy', prec=200)
0.006591139357460719
sage: airy_ai(3).n(algorithm='whatever')
0.00659113935746072

And I don't like this either (parent should be RR):

sage: parent(airy_ai(3).n(algorithm='scipy'))
Real Double Field

[jdemeyer]

What's the point of

if len(args) > 0:
   raise TypeError(("symbolic function airy_ai takes at most 3 arguments "
      "({} given)").format(len(args) + 3))

why not simply remove *args completely then?

[rws]

Replying to jdemeyer:

And why the change to def hypergeometric_U? The new code seems quite hackish, why is it needed and how does it relate to this ticket?

I think that unrelated patch can be safely removed because scipy evaluation is replaced by mpmath in #14896 (of course hoping that that gets reviewed at some time).

[git]

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

​15e7e34	Merge branch 'develop' into t/12455/public/ticket/12455
​2c0931c	12455: remove fishy code in hypergeometric_U()
​b4213d3	12455: with algorithm=scipy return reals for real argument
​b0841c9	12455: catch attempts to exceed precision with algorithm=scipy
​484a615	12455: catch unknown evalf algorithms
​58e006c	12455: remove superfluous args and check
​870e003	12455: fix plot doctests
​61847c4	Merge branch 'public/ticket/12455' of trac.sagemath.org:sage into t/12455/public/ticket/12455

[git]

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

​6d10729	Simplify numerical evaluation of BuiltinFunctions
​b6e1ed4	Merge remote-tracking branches 'origin/u/jdemeyer/ticket/17131' and 'origin/u/jdemeyer/ticket/17133' into ticket/17130
​382f97a	Merge branch 'u/jdemeyer/ticket/17130' of trac.sagemath.org:sage into 6.5beta1
​7265989	17130: reviewer's patch: fix typo
​c47dbd4	Fix Trac #17328 again in a better way
​a486db2	Call the factorial() method of an Integer
​9d3cbbd	Fix numerical noise
​abab222	Fix more numerical noise
​01614f7	Merge branch 'u/jdemeyer/ticket/17130' of trac.sagemath.org:sage into t/12455/public/ticket/12455
​aac86a2	12455: simplifications due to trac 17130

[kcrisman]

Frédéric, how much of this did you positively review up to now? In principle you and rws can review each other's contributions here. It would take me a while to get up to speed here.

[chapoton]

a typo in line 5 of the doc:

Airy functions are solutions to the differential equation `f''(z) +f(z)x=0`.

The differential equation is wrong, should be f'' - x f = 0

[git]

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

​3cac775	Merge branch 'develop' into t/12455/public/ticket/12455
​070e728	12455: doc typos

[mmezzarobba]

x should be z in the differential equation

[mmezzarobba]

What exactly still needs review here?

The code looks pretty good to me overall, but there are lots of things I don't understand in detail. (For a simple example, what is the motivation for splitting the implementation in so many unrelated classes?) And yet I'm tempted to give this ticket positive review (leaving possible remaining issues for later), since many other people have looked at it and it is clearly an improvement over what sage currently has.

[mmezzarobba]

More nitpicking, not necessarily for this ticket:

• Raising a ValueError in FunctionAiryAiGeneral._derivative_ when diff_param == 0 may not be the most appropriate, since (if I'm not mistaken) the partial derivative would make sense mathematically.
• Is it necessary to special-case alpha == 0 etc. in FunctionAiryAiGeneral._evalf_?

[mmezzarobba]

Replying to mmezzarobba:

• Is it necessary to special-case alpha == 0 etc. in FunctionAiryAiGeneral._evalf_?

Scratch that, I misread the code.

[git]

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

​91530d0

Ai/Bi: doc fixes and clarifications

[mmezzarobba]

One last thing: IMO hold_derivative should be True by default in airy_ai and airy_bi, except perhaps when the differentiation order is one. But I don't know if that's consistent with the way things are done in other parts of sage symbolics. Thoughts?

[rws]

Replying to mmezzarobba:

x should be z in the differential equation

Strange typo.

• Raising a ValueError in FunctionAiryAiGeneral._derivative_ when diff_param == 0 may not be the most appropriate, since (if I'm not mistaken) the partial derivative would make sense mathematically.

A quick look at old and new code shows 1. ValueError, 2. NotImplementedError, 3. assert. So, this needs to be unified to NotImplementedError.

One last thing: IMO hold_derivative should be True by default in airy_ai and airy_bi, except perhaps when the differentiation order is one. But I don't know if that's consistent with the way things are done in other parts of sage symbolics. Thoughts?

I believe the code of this ticket is the only one that has such a parameter, so please feel free to improve it.

I have looked at your patch and it's fine, so if you think the rest is OK please set positive.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​347cd08	#12455 Airy functions: doc fixes and clarifications
​6aaf2da	#12455 Airy funs: raise NotImplementedError on diff(airy_?i(α,x),α)
​c80be63	#12455 Airy funs: hold_derivative by default
​2f6945a	#12455 Airy functions: avoid calling deprecated function

[mmezzarobba]

Replying to rws:

• Raising a ValueError in FunctionAiryAiGeneral._derivative_ when diff_param == 0 may not be the most appropriate, since (if I'm not mistaken) the partial derivative would make sense mathematically.

A quick look at old and new code shows 1. ValueError, 2. NotImplementedError, 3. assert. So, this needs to be unified to NotImplementedError.

One last thing: IMO hold_derivative should be True by default in airy_ai and airy_bi, except perhaps when the differentiation order is one. But I don't know if that's consistent with the way things are done in other parts of sage symbolics. Thoughts?

I believe the code of this ticket is the only one that has such a parameter, so please feel free to improve it.

Both done (now rather than later to avoid breaking the interface).

I have looked at your patch and it's fine, so if you think the rest is OK please set positive.

I'm okay with everything prior to my commits. Could you (or someone else) have a quick look at my last changes and set the ticket to positive review? Thanks!

[rws]

Is fine and passes tests in functions/. Thank you too.