Opened 11 years ago
Last modified 8 years ago
#11143 closed defect
Add various Maxima special functions to symbol table — at Version 11
Reported by: | kcrisman | Owned by: | burcin |
---|---|---|---|
Priority: | major | Milestone: | sage-5.3 |
Component: | symbolics | Keywords: | ei Ei special function maxima sd32 sd40.5 |
Cc: | Merged in: | ||
Authors: | Reviewers: | ||
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
Description (last modified by )
We're missing some conversions from Maxima. Like exponential integrals of various kinds.
sage: f(x) = e^(-x) * log(x+1) sage: uu = integral(f,x,0,oo) sage: uu x |--> e*expintegral_e(1, 1)
See this ask.sagemath post for some details.
Current symbol conversion table
From sage.symbolic.pynac.symbol_table['maxima']
as of Sage-4.7
Maxima ---> Sage %gamma ---> euler_gamma %pi ---> pi (1+sqrt(5))/2 ---> golden_ratio acos ---> arccos acosh ---> arccosh acot ---> arccot acoth ---> arccoth acsc ---> arccsc acsch ---> arccsch asec ---> arcsec asech ---> arcsech asin ---> arcsin asinh ---> arcsinh atan ---> arctan atan2 ---> arctan2 atanh ---> arctanh binomial ---> binomial brun ---> brun catalan ---> catalan ceiling ---> ceil cos ---> cos delta ---> dirac_delta elliptic_e ---> elliptic_e elliptic_ec ---> elliptic_ec elliptic_eu ---> elliptic_eu elliptic_f ---> elliptic_f elliptic_kc ---> elliptic_kc elliptic_pi ---> elliptic_pi exp ---> exp expintegral_e ---> En factorial ---> factorial gamma_incomplete ---> gamma glaisher ---> glaisher imagpart ---> imag_part inf ---> +Infinity infinity ---> Infinity khinchin ---> khinchin kron_delta ---> kronecker_delta li[2] ---> dilog log ---> log log(2) ---> log2 mertens ---> mertens minf ---> -Infinity psi[0] ---> psi realpart ---> real_part signum ---> sgn sin ---> sin twinprime ---> twinprime
Summary of missing conversions
Special functions defined in Maxima
(http://maxima.sourceforge.net/docs/manual/en/maxima_16.html#SEC56)
bessel_j (index, expr) Bessel function, 1st kind bessel_y (index, expr) Bessel function, 2nd kind bessel_i (index, expr) Modified Bessel function, 1st kind bessel_k (index, expr) Modified Bessel function, 2nd kind
- Notes: bessel_I, bessel_J, etc. are functions in Sage for numerical evaluation. There is also the
Bessel
class, but no conversions from Maxima's bessel_i etc. to Sage.
hankel_1 (v,z) Hankel function of the 1st kind hankel_2 (v,z) Hankel function of the 2nd kind struve_h (v,z) Struve H function struve_l (v,z) Struve L function
- Notes: None of these functions are currently exposed at the top level in Sage. Evaluation is possible using mpmath.
assoc_legendre_p[v,u] (z) Legendre function of degree v and order u assoc_legendre_q[v,u] (z) Legendre function, 2nd kind
- Notes: In Sage we have
legendre_P(n, x)
andlegendre_Q(n, x)
both described as Legendre functions. It's not clear to me how there are related to Maxima's versions since the number of arguments differs.
%f[p,q] ([], [], expr) Generalized Hypergeometric function hypergeometric(l1, l2, z) Hypergeometric function slommel %m[u,k] (z) Whittaker function, 1st kind %w[u,k] (z) Whittaker function, 2nd kind
- Notes:
hypergeometric(l1, l2, z)
needs a conversion to Sage'shypergeometric_U
. The others can be evaluated using mpmath.slommel
is presumably mpmath'slommels1()
orlommels2()
(or both?). This isn't well documented in Maxima.
expintegral_e (v,z) Exponential integral E expintegral_e1 (z) Exponential integral E1 expintegral_ei (z) Exponential integral Ei expintegral_li (z) Logarithmic integral Li expintegral_si (z) Exponential integral Si expintegral_ci (z) Exponential integral Ci expintegral_shi (z) Exponential integral Shi expintegral_chi (z) Exponential integral Chi erfc (z) Complement of the erf function
- Notes: The exponential integral functions
expintegral_e1
andexpintegral_ei (z)
are calledexponential_integral_1
andEi
resp. in Sage. They both need conversions. The rest needBuiltinFunction
classes defined for them with evaluation handled by mpmath and the symbol table conversion added. Also,erfc
is callederror_fcn
, so also needs a conversion.
kelliptic (z) Complete elliptic integral of the first kind (K) parabolic_cylinder_d (v,z) Parabolic cylinder D function
- Notes:
kelliptic(z)
needs a conversion toelliptic_kc
in Sage andparabolic_cylinder_d (v,z)
does not seem to be exposed at top level. It can be evaluated by mpmath.
Change History (11)
comment:1 follow-ups: ↓ 3 ↓ 4 Changed 11 years ago by
comment:2 Changed 11 years ago by
- Cc benjaminfjones added
comment:3 in reply to: ↑ 1 Changed 11 years ago by
Replying to benjaminfjones:
As far as I can tell, the general
exponential_e
function isn't available directly in Sage or in PARI (which is used to evaluate theexponential_integral_1
function in Sage).
That's unfortunate. However, mpmath seems to have it. So we could create a symbolic version and have the _eval_
method call mpmath, which we seem to be moving to.
Also, it's possible to get maxima to rewrite the exponential integrals in terms of gamma functions like so:
sage: maxima.eval('expintrep:gamma_incomplete') 'gamma_incomplete' sage: maxima.integrate(exp(-x)*log(x+1), x, 0, oo) %e*gamma_incomplete(0,1) sage: N(e*gamma(0,1), digits=18) 0.596347362323194074
Interesting.
But as you see,
gamma_incomplete
isn't defined in Sage either, but the tablesage.symbolic.pynac.symbol_table['maxima']
lists the Sage equivalentgamma
.
That should be ok; the whole point of the table is to convert into the Sage equivalent.
By the way, the owner on the ticket is @burcin, does that mean they are working on it currently?
No, that is an automatic thing that happens. It is possible to be designated an 'owner' of a ticket in a given component, which basically means you automatically get updates. If you want to 'own' it, please do! We really have plenty of special functions in Sage, but they are not always well exposed at the top level.
Incidentally, once you comment on a ticket, I believe the default is to copy you in on all replies. So you didn't have to cc: yourself specially :)
comment:4 in reply to: ↑ 1 Changed 11 years ago by
Replying to benjaminfjones:
But as you see,
gamma_incomplete
isn't defined in Sage either, but the tablesage.symbolic.pynac.symbol_table['maxima']
lists the Sage equivalentgamma
. Anyway, it should be possible to have the maxima interface (with the help of maxima itself) rewrite any exponential integral that Sage doesn't have in terms of gamma functions.
Incomplete gamma is defined in Sage. You can access it directly though incomplete_gamma()
or gamma_inc()
. The top level function gamma()
behaves like incomplete gamma if you give it two arguments. IIRC, this is similar to maple.
By the way, the owner on the ticket is @burcin, does that mean they are working on it currently?
I am not working on it. The ticket status assigned
is supposed to indicate that the owner is working on the problem, but we don't use that much either.
comment:5 follow-up: ↓ 6 Changed 11 years ago by
I guess the point of this ticket is to define symbolic function in Sage to represent exponential integrals, etc. The symbolic function class handles adding stuff to the conversion table automatically.
Can we replace this ticket with several beginner tickets? One for each function we are missing.
comment:6 in reply to: ↑ 5 Changed 11 years ago by
Replying to burcin:
I guess the point of this ticket is to define symbolic function in Sage to represent exponential integrals, etc. The symbolic function class handles adding stuff to the conversion table automatically.
Can we replace this ticket with several beginner tickets? One for each function we are missing.
Well, that would be nice. But we could also presumably do it directly, if that would solve this problem for now. Well, either is fine as long as it were to happen. If you do split this, be sure to give a good template (I mean a link to the template, not write it yourself).
comment:7 Changed 11 years ago by
I'm attempting to write a template for the expintegral_e
function (denoted E_n(z) in A&S). As I'm looking through the code, I see several models used for the functions and classes in sage/functions/special.py
and sage/functions/transcendental.py
- Functions like
Function_exp_integral
(also calledEi
) are defined as classes that inherit fromBuiltInFunction
and call the mpmath implementation when evaluated. The functionDickmanRho
also does this and includes other nice methods for approximating values and power series. - Functions like
EllipticE
inherit fromMaximaFunction
which handles evaluation, etc. through Maxima. It seems there is an advantage to the mpmath implementation because presumably the interface is faster and the precision is arbitrary (whereas Maxima is limited to 53 bits). - Functions like
Li
anderror_fcn
are simply wrapper functions that try to evaluate the input symbolically or numerically depending on context. - In
sage/functions/trig.py
there is a mixture of classes that derive fromGinacFunction
(and include information in their__init__
methods about conversions to other systems like Maxima or Mathematica) and also functions that derive fromBuiltinFunction
. It's not clear to me why some functions are Ginac and some are Builtin.
Questions:
- For the purposes of this ticket, what is recommended? @kcrisman 's comment leads me to believe that inheriting from the
BuiltinFunction
class and usingmpmath
for evaluation is preferable. - Where should the various exponential integral special functions that we are missing go?
sage/functions/special.py
,sage/functions/transcendental.py
, or somewhere else?
comment:8 Changed 11 years ago by
Those are good questions.
I think the most important thing is to make sure that whatever is implemented has
- good numerical evaluation (perhaps via mpmath)
- translates back and forth to Maxima properly (for integration and limits)
My sense is that BuiltinFunction would be best. GinacFunction probably is only good for things that in fact evaluate in Ginac. This explains trig.py. Ginac page shows that the ones which are GinacFunctions are exactly the ones which Ginac in fact has. I don't think it has most of these other functions.
As for MaximaFunction, it seems to inherit from BuiltinFunction and lives in the special functions file. This dates from the days when Sage had very few options for symbolic stuff and evaluation, and so it just does a few extra things. If we moved to mpmath for a given function, we would probably use BuiltinFunction and then add evaluation options for Maxima and add to the conversion dictionary as needed.
In fact, it wouldn't be a bad idea to have two different eval procedures if possible...
As for where such things go, probably it would make sense to separate a lot of these special functions out into a separate file. The distinction between transcendental and special is not totally obvious, for instance :)
By the way, we certainly want fewer of the wrapper functions! But that will take a lot of tedious work.
comment:9 Changed 11 years ago by
By the way, adding this will really be a great help. Sage has so many of the things almost anybody needs, but if you have to use mpmath or GSL or R or something else directly, it sort of makes Sage a moot point. The idea is having a one-stop shop.
comment:10 Changed 11 years ago by
I've uploaded a first shot at a template for the functions we want to add in this ticket. The patch exposes the general complex exponential integral function En
also called Function_exp_integral_En
by adding a class to sage/functions/special.py
(I can change where it goes later on if needed / desired). Numerical evaluation is handled by mpmath and symbolics are handled by Sage (e.g. the derivative) and Maxima (e.g. the antiderivative).
One of the docstring examples shows that the integral of e^(-x) * log(x+1)
from the ticket description is now evaluated properly.
Any comments or suggestions?
comment:11 Changed 10 years ago by
- Description modified (diff)
- Keywords special function maxima added
As far as I can tell, the general
exponential_e
function isn't available directly in Sage or in PARI (which is used to evaluate theexponential_integral_1
function in Sage).Also, it's possible to get maxima to rewrite the exponential integrals in terms of gamma functions like so:
But as you see,
gamma_incomplete
isn't defined in Sage either, but the tablesage.symbolic.pynac.symbol_table['maxima']
lists the Sage equivalentgamma
. Anyway, it should be possible to have the maxima interface (with the help of maxima itself) rewrite any exponential integral that Sage doesn't have in terms of gamma functions.By the way, the owner on the ticket is @burcin, does that mean they are working on it currently?