Make incomplete gamma function symbolic

[kcrisman]

We don't appear to have this, though perhaps it is in Pynac/Ginac? already. We also need to translate Maxima's gamma_incomplete to our gamma_inc or incomplete_gamma as part of this ticket.

[burcin]

After a quick look at the GiNaC sources, I don't think there is an implementation of the incomplete gamma function in there.

Numerical evaluation of this can be done with the gammainc() function of mpmath.

The conversion should just work, if the conversions dictionary argument of BuiltinFunction.__init__ contains maxima='gamma_incomplete'.

[kcrisman]

Replying to burcin:

After a quick look at the GiNaC sources, I don't think there is an implementation of the incomplete gamma function in there.

Okay.

Numerical evaluation of this can be done with the gammainc() function of mpmath.

But will it give us symbolic evaluation ala gamma_inc(1,1)=1/e? That would be best.

The conversion should just work, if the conversions dictionary argument of BuiltinFunction.__init__ contains maxima='gamma_incomplete'.

Yes, of course, but we still have to do it :)

[was]

Numerical evaluation of this can be done with the gammainc() function of mpmath.

Doesn't PARI also have an incomplete gamma implementation (it's used by RR and CC, I think)?

[fstan]

Make Ei symbolic.

[fstan]

Make incomplete gamma function symbolic.

[fstan]

The 2 patches should make the incomplete gamma function symbolic. The exponential integral Ei was used in its symbolic evaluation, therefore it was also made symbolic.

[zimmerma]

I tried the exponential integral patch and it works ok. However I don't understand the changes in random_tests.py. Also the following could be evaluated:

sage: diff(Ei(x),x)
D[0](Ei)(x)
sage: integrate(Ei(x),x)
integrate(Ei(x), x)

Should this be in a separate ticket?

[kcrisman]

Replying to zimmerma:

I tried the exponential integral patch and it works ok. However I don't understand the changes in random_tests.py.

This makes a 'random' symbolic expression, and when we add new symbolic functions (and sometimes when we change them) this doctest needs to be changed, not because anything bad happened.

[fstan]

Replying to zimmerma:

I tried the exponential integral patch and it works ok. However I don't understand the changes in random_tests.py. Also the following could be evaluated:

sage: diff(Ei(x),x)
D[0](Ei)(x)
sage: integrate(Ei(x),x)
integrate(Ei(x), x)

Should this be in a separate ticket?

Thank you for the review. I'll include the derivative and submit again. Integration looks more complicated.

Greetings,
Flavia

[zimmerma]

I tried sage -t * with both patches, and I get one additional failure with respect to those I get due to #7773:

tarte% sage -t "rings/complex_number.pyx"
sage -t  "rings/complex_number.pyx"                         
**********************************************************************
File "/usr/local/sage-4.3-core2/devel/sage-main/sage/rings/complex_number.pyx", line 1611:
    sage: gamma_inc(2, 5)
Expected:
    0.0404276819945128
Got:
    gamma(2, 5)

Consider also the following:

sage: gamma_inc(2, 5.)
0.0404276819945128
sage: gamma(2, 5) 
1

[burcin]

Replying to zimmerma:

Consider also the following:

sage: gamma_inc(2, 5.)
0.0404276819945128
sage: gamma(2, 5) 
1

We should write a python function wrapping gamma_inc and gamma, similar to the provided for psi() in #6961.

The fact that gamma() accepts two arguments at the moment is a bug that I introduced in #7490. Looking at the code, it also doesn't handle the prec parameter correctly. I'll upload a patch with a wrapper gamma() function and a fix for the prec issue.

Replying to fstan:

Integration looks more complicated.

Integration cannot be handled with a custom method in symbolic functions. The patches attached to #6465 should make this easier.

[zimmerma]

Flavia, please can you fix the rings/complex_number.pyx issue?

[fstan]

Added derivative method.

[fstan]

Fixed tests from complex_number.pyx

[fstan]

I've uploaded new patches which should fix the docs and the derivative.

Greetings,
Flavia

[burcin]

rebased to 4.3.3.alpha1

[burcin]

rebased to 4.3.3.alpha1

[burcin]

wrapper for gamma and incomplete gamma

[burcin]

wrapper for gamma and incomplete gamma

[burcin]

attachment:trac_7748-gamma_wrapper.2.patch​ adds a new wrapper function to call gamma or incomplete gamma depending on the number of arguments. It also corrects the way prec parameter to gamma was handled.

Flavia's patches needed to be rebased to 4.3.3.alpha1 since the changes to random_tests.py didn't apply anymore.

The patches to be applied are (in this order):

• attachment:trac_7748-exp_integral_ver2.4.3.3.alpha1.patch​
• attachment:trac_7748-incomplete_gamma_ver2.4.3.3.alpha1.patch​
• attachment:trac_7748-gamma_wrapper.2.patch​

I'm changing this to needs review. :)

[zimmerma]

a partial review against 4.3.3:

sage -t  "devel/sage-main/sage/schemes/elliptic_curves/ell_rational_field.py"
...
    sage: E.Lambda(1.4+0.5*I, 50)
      File "/usr/local/sage-4.3.3/sage/local/lib/python/site-packages/sage/schemes/elliptic_curves/ell_rational_field.py", line 3088, in Lambda
        Gamma_inc = transcendental.gamma_inc
    AttributeError: 'module' object has no attribute 'gamma_inc'

This test was ok with vanilla 4.3.3.

[zimmerma]

apart from the above failure, all tests pass with sage 4.3.3 (apart from the Fedora 12 Segfaults).

[burcin]

fix new doctest failure

[burcin]

I added a fix for the doctest failure in comment:13 to my patch, attachment:trac_7748-gamma_wrapper.3.patch​.

The patches to be applied are (in this order):

• attachment:trac_7748-exp_integral_ver2.4.3.3.alpha1.patch​
• attachment:trac_7748-incomplete_gamma_ver2.4.3.3.alpha1.patch​
• attachment:trac_7748-gamma_wrapper.3.patch​

I am OK with Flavia's changes. If someone can review my patch this can still get in 4.3.4. :)

[zimmerma]

Everything works now. Great work!

Paul

[burcin]

Hi Minh,

What is holding up this ticket? Note that it has a positive review and the patches to be applied are listed in comment:15.

Cheers,
Burcin

[mvngu]

Replying to burcin:

What is holding up this ticket?

I can't merge this ticket. We are now in feature freeze.

[jhpalmieri]

Merged in 4.4.alpha0:

• trac_7748-exp_integral_ver2.4.3.3.alpha1.patch
• trac_7748-incomplete_gamma_ver2.4.3.3.alpha1.patch
• trac_7748-gamma_wrapper.3.patch