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                                 Traceback (most recent call last)

/home/oscar/Escritorio/tesis/calculos/<ipython console> in <module>()

/home/oscar/sage/sage-4.7.1/local/lib/python2.6/site-packages/sage/functions/special.pyc in airy_ai(x)
    621    """
    622    _init()
--> 623    return RDF(meval("airy_ai(%s)"%RDF(x)))
    624 
    625 def airy_bi(x):

/home/oscar/sage/sage-4.7.1/local/lib/python2.6/site-packages/sage/structure/parent.so in sage.structure.parent.Parent.__call__ (sage/structure/parent.c:7102)()

/home/oscar/sage/sage-4.7.1/local/lib/python2.6/site-packages/sage/structure/coerce_maps.so in sage.structure.coerce_maps.NamedConvertMap._call_ (sage/structure/coerce_maps.c:4221)()

/home/oscar/sage/sage-4.7.1/local/lib/python2.6/site-packages/sage/symbolic/expression.so in sage.symbolic.expression.Expression._real_double_ (sage/symbolic/expression.cpp:5391)()

/home/oscar/sage/sage-4.7.1/local/lib/python2.6/site-packages/sage/symbolic/expression.so in sage.symbolic.expression.Expression._eval_self (sage/symbolic/expression.cpp:4898)()

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.

---

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

[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

Patchbot apply trac_12455_newstyle_airy.patch trac_12455_newstyle_airy2.patch

[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