define symbolic functions for exponential integrals

[kcrisman]

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) and legendre_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's hypergeometric_U. The others can be evaluated using mpmath. slommel is presumably mpmath's lommels1() or lommels2() (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 and expintegral_ei (z) are called exponential_integral_1 and Ei resp. in Sage. They both need conversions. The rest need BuiltinFunction classes defined for them with evaluation handled by mpmath and the symbol table conversion added. Also, erfc is called error_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 to elliptic_kc in Sage and parabolic_cylinder_d (v,z) does not seem to be exposed at top level. It can be evaluated by mpmath.

---

Apply to the Sage library:

1. attachment: trac_11143-v2.5-rebased.4.patch
2. attachment: trac-11143-ref.2.patch
3. attachment: trac_11143-pynac-serials.patch

Depends on #13109

Component: symbolics

Keywords: ei Ei special function maxima sd32 sd40.5

Author: Benjamin Jones, Volker Braun

Reviewer: Burcin Erocal, Karl-Dieter Crisman, William Stein

Merged: sage-5.3.beta2

Issue created by migration from https://trac.sagemath.org/ticket/11143

[benjaminfjones]

comment:1

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 the exponential_integral_1 function in Sage).

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

But as you see, gamma_incomplete isn't defined in Sage either, but the table sage.symbolic.pynac.symbol_table['maxima'] lists the Sage equivalent gamma. 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?

[kcrisman]

comment:3

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 the exponential_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 table sage.symbolic.pynac.symbol_table['maxima'] lists the Sage equivalent gamma.

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 :)

[burcin]

comment:4

Replying to @benjaminfjones:

But as you see, gamma_incomplete isn't defined in Sage either, but the table sage.symbolic.pynac.symbol_table['maxima'] lists the Sage equivalent gamma. 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.

[burcin]

comment:5

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.

[kcrisman]

comment:6

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).

[benjaminfjones]

comment:7

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 called Ei) are defined as classes that inherit from BuiltInFunction and call the mpmath implementation when evaluated. The function DickmanRho also does this and includes other nice methods for approximating values and power series.
• Functions like EllipticE inherit from MaximaFunction 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 and error_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 from GinacFunction (and include information in their __init__ methods about conversions to other systems like Maxima or Mathematica) and also functions that derive from BuiltinFunction. 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 using mpmath 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?

[kcrisman]

comment:8

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. This 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.

[kcrisman]

comment:9

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.

[benjaminfjones]

comment:10

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?

[benjaminfjones]

Changed keywords from ei Ei to ei Ei special function maxima

[burcin]

comment:12

The patch looks great. Thanks for the template. A few suggestions:

• The function name should be more explicit. I suggest exp_integral_e.
• In the top level name space En can be an alias for this function (though I'd prefer not to take up a two letter name), but we should have the long name available. It is easier to find all these functions by exp_integral<tab> than E<tab>.
• See _eval_ method of sage.functions.other.Function_gamma_inc for an example of how to find a common parent for the arguments
• The call to mpmath does not require the prec parameter any more. If you pass a Sage element to mpmath it handles the precision correctly. You can delete the code for this in _evalf_().
• You should remove the __call__() method altogether. The only purpose of that is to display the deprecation notice. Since you are implementing a new function here, there is nothing being deprecated.
• In _derivative_() the call to this function should be exp_integral_e(n-1, z), assuming you change the function name accordingly.

I changed the ticket description to limit this to implementing symbolic functions for exponential integrals. We can use the wiki page for a general overview of the progress on symbolic functions.

[burcin]

Author: Benjamin Jones

[burcin]

Reviewer: Burcin Erocal

[benjaminfjones]

comment:13

Thanks for the feedback, Burcin! I feel like I'm developing an understanding for how symbolic functions are handled in Sage now.

To your suggestions:

• I agree, the changed the name of the class to Function_exp_integral_e and the global function name to exp_integral_e. I think it would make sense as part of this ticket to add an alias of exp_integral_1 for the existing exponential_integral_1 to be consistent.
• Good point, I didn't think of that.
• The function now correctly coerces the two arguments to the same parent. I also added automatic evaluation of some special cases (n=0 or z=0) as in the gamma_inc function.
• I played around with removing the prec argument, but didn't get the results I expected. I found that passing the parent explicitly works. The common parent is determined in _eval_ (as in the gamma_inc function).
• Done.
• Done.

Let me know what you think. Once we've agreed on a good template for this one function, I will implement the rest of the exponential integral functions that are now listed on the wiki and upload to this ticket.

[burcin]

comment:14

Perfect! I can only nitpick about documentation. :)

• lines should be < 80 characters long. This helps when viewing help in a terminal.
• I don't know how testing for 0 in _eval_ effects performance. This is a hard problem. See the _eval_ methods in sage/functions/generalized.py for a workaround if this is too slow.

You're right, we should add an alias exp_integral_1.

Thanks for working on this.

[benjaminfjones]

comment:15

The tests similar to if z == 0 in _eval_ do make a big difference. I guess this is well known, but I didn't realize how big the speed difference is.

I made a little table below of some timings. In the first table , _eval_ includes tests if z == 0 and n > 1 and if n == 0. In the second table, there are no such if statements (so the special cases are not implemented). In the third table, the special cases are implementes as in sage/functions/generalized.py where an approximation of z (or n) is produced and checked instead of the symbol.

with ``if z == 0``

=============================================	========
  Test											 Time
=============================================	========
sage: timeit("f = exp_integral_e(n,z)")			 1.44 ms
sage: timeit("f = exp_integral_e(n,0)")			 929 µs
sage: timeit("f = exp_integral_e(0,z)")			 1.41 ms
sage: timeit("f = exp_integral_e(1.0,1.0)")		 158 µs
=============================================	========

without ``if z == 0``

=============================================	======
  Test											 Time
=============================================	======
sage: timeit("f = exp_integral_e(n,z)")			541 µs
sage: timeit("f = exp_integral_e(n,0)")			300 µs
sage: timeit("f = exp_integral_e(0,z)")			299 µs
sage: timeit("f = exp_integral_e(1.0,1.0)")		161 µs
=============================================	======

with:

.. code-block:: python

	try:
	    approx_z = ComplexIntervalField()(z)
	    # if z is zero and n > 1
	    if bool(approx_z.imag() == 0) and bool(approx_z.real() == 0):
	        if n > 1:
	            return 1/(n-1)
	except: # z is symbolic
	    pass
	# if n is zero
	try:
	    approx_n = ComplexIntervalField()(n)
	    if bool(approx_n.imag() == 0) and bool(approx_n.real() == 0):
	        return exp(-z)/z
	except: # n is symbolic
	    pass

=============================================	======
  Test											 Time
=============================================	======
sage: timeit("f = exp_integral_e(n,z)")			570 µs
sage: timeit("f = exp_integral_e(n,0)")			576 µs
sage: timeit("f = exp_integral_e(0,z)")			1.05 ms
sage: timeit("f = exp_integral_e(1.0,1.0)")		160 µs
=============================================	======

Timings in tables 2 and 3 are close except in the case where exp(-z)/z is
returned, whereas table 1 is anywhere from a factor of 3 to 5 slower than in table 2 when a symbolic argument is passed. Anyway, I thought I'd include the above for other beginners such as myself.

---

Another thing I discovered is that these two special cases that I was
implementing are known to Maxima:

sage: f = exp_integral_e(0,x)
sage: f
exp_integral_e(0,x)
sage: f.simplify()
e^(-x)/x

sage: nn = var('nn')
sage: assume(nn > 1)
sage: f = exp_integral_e(nn, 0)
sage: f
exp_integral_e(nn, 0)
sage: f.simplify()
1/(nn - 1)

So I think in the interest of speedy evaluation it's best to leave the special
cases out, but point out in the documentation that Maxima knows about them.

I've uploaded a new patch. I'll start implementing the other exponential integrals using this as a template.

[burcin]

comment:16

The timings are really instructive, we should link to them from wiki page. Thank you for this careful study.

Replying to @benjaminfjones:

So I think in the interest of speedy evaluation it's best to leave the special
cases out, but point out in the documentation that Maxima knows about them.

I think your timings show that using the CIF approximation is not such a big penalty. Note that your timings also reflect the construction of the result and not just including that change. It is better if Sage can do the evaluation automatically without having to pass through a simplify() call. I'd like them in, with the approximation approach.

Since this approximation is being called so often, we should make it a method of symbolic expressions. I opened a new ticket for this: #11513. This ticket should depend on that.

I will provide a preliminary patch for #11513 soon.

[burcin]

comment:17

I attached a very simple patch to #11513. Now we can write the check for zero as:

def is_zero(z):
    if isinstance(z, Expression):
        return z._is_numerically_zero()
    else:
        return not z

It would be nice if we could simplify this further, but I need to go and do other things now. :)

[benjaminfjones]

comment:18

The patch now depends on #11513

Here are new timings for the _eval_ method with _is_numerically_zero():

.. code-block:: python

    # special case: *quickly* test if (z == 0 and n > 1)
    if isinstance(z, Expression):
        if z._is_numerically_zero():
            z_zero = True # for later
            if n > 1:
                return 1/(n-1)
        else:
            if not z: 
                z_zero = True
                if n > 1:
                    return 1/(n-1)

======================================================  =======
  Test                                                   Time
======================================================  =======
sage: timeit("f = exp_integral_e(n,z)")                 535 µs
sage: timeit("f = exp_integral_e(n,0)")                 482 µs
sage: assume(n > 1); timeit("f = exp_integral_e(n,0)")  3.56 ms
sage: timeit("f = exp_integral_e(0,z)")                 968 µs
sage: timeit("f = exp_integral_e(1.0,1.0)")             160 µs
======================================================  =======

I realized that in row 2 of the previous timings I neglected to assume n > 1 so those timings aren't giving much information since the expression is left unevaluated like in row 1. The new row 3 includes that assumption so that the simplified result 1/(n-1) is created and returned.

I'll update the timings above and move these tables to the wiki.

[benjaminfjones]

Attachment: trac_11143_En.patch.gz

added the symbolic function exp_integral_e

[benjaminfjones]

comment:19

In the patch, I'm added a function to sage/functions/special.py. But with the six new functions, this is going to be a lot of code in the one file.

I was thinking of moving the six new functions to a new file sage/functions/exp_integral.py. Any thoughts? If that seems like a good idea, does it also make sense to move the exp integrals that already exist in Sage to the same place (e.g. exponential_integral_1 lives now in sage/functions/transcendental.py.

[jdemeyer]

comment:88

There is a trivial doctest failure on 32-bit i386 systems (possibly other systems too):

sage -t  --long -force_lib devel/sage/sage/functions/exp_integral.py
**********************************************************************
File "/var/lib/buildbot/build/sage/arando-1/arando_full/build/sage-5.3.beta1/devel/sage-main/sage/functions/exp_integral.py", line 1297:
    sage: w = exponential_integral_1(2,4); w
Expected:
    [0.04890051070806112, 0.0037793524098489067, 0.00036008245216265873, 3.7665622843924751e-05] 
Got:
    [0.04890051070806112, 0.0037793524098489067, 0.00036008245216265873, 3.766562284392475e-05]
**********************************************************************

[benjaminfjones]

Attachment: trac_11143-v2.5-rebased.3.patch.gz

fixed failing doctest

[benjaminfjones]

comment:89

Fixed the failing doctest (for the reviewer: this only changed lines 1300--1301 in sage/functions/exp_integral.py).

[jdemeyer]

comment:90

Wouldn't a relative tolerance be better than an absolute one? With the absolute tolerance, the fourth value can essentially be anything.

[benjaminfjones]

comment:91

Yes, you're right. I'll change that.

[benjaminfjones]

changed abs tol to rel tol

[benjaminfjones]

comment:92

Attachment: trac_11143-v2.5-rebased.4.patch.gz

[benjaminfjones]

comment:93

Patchbot, please kindly apply:

• attachment: trac_11143-v2.5-rebased.4.patch
• attachment: trac-11143-ref.2.patch
• attachment: trac_11143-pynac-serials.patch
  to the Sage library.

Thank you in advance.

[jdemeyer]

comment:94

I'm happy with the changes to that line. I haven't re-checked the whole patch again, but I thrust that you didn't change anything else.

[jdemeyer]

Merged: sage-5.3.beta2

[kcrisman]

comment:97

See #14766 for a followup.