implement symbolic lower incomplete gamma function #16697

rwst · 2014-07-22T08:45:10Z

This is actually a defect because we leave a result from Maxima undefined:

sage: hypergeometric([1],[b],x).simplify_hypergeometric()
(b - 1)*x^(-b + 1)*e^x*gamma_greek(b - 1, x)
sage: gamma_greek
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-18-6e90901bc5cb> in <module>()
----> 1 gamma_greek

NameError: name 'gamma_greek' is not defined

https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function

Mathematica seems to have Gamma[a,z] for upper and Gamma[a,0,z] for lower; Maple seems to have upper Gamma. gamma_inc (the upper one in Sage) gets immediately converted to gamma(a,x). The symbolic functions gamma_inc==incomplete_gamma are converted and never returned to the user as expression:

sage: gamma_inc(x,x,hold=True)
gamma(x, x)
sage: incomplete_gamma(x,x,hold=True)
gamma(x, x)
sage: assume(x>0)
sage: integral(t^(s-1)*e^(-t),t,0,x)
-gamma(s, x) + gamma(s)

This ticket should deprecate "incomplete_gamma" and add the symbolic function gamma_inc_lower, leaving open the question of the global alias for and the displayed name of Function_gamma_inc.

Previous part of description:

Provide all three "user input interfaces" gamma_inc, incomplete_gamma and lower_incomplete_gamma, and convert the Maxima gamma_greek to -gamma(a, x) + gamma(a)
Provide all three "user input interfaces" gamma_inc, incomplete_gamma and lower_incomplete_gamma, and convert to gamma(a,x) and gamma(a,0,x)n on output
Provide all three "user input interfaces" gamma_inc, incomplete_gamma and lower_incomplete_gamma, and have incomplete_gamma and lower_incomplete_gamma as result instead of `gamma(...)'
Change the user interface completely to lower_incomplete_gamma and upper_incomplete_gamma
Change the user interface completely to gamma_inc_lower and gamma_inc_upper
Change the user interface completely to gamma(a,x) and gamma(a,0,x)

Depends on #20185

CC: @kcrisman @nbruin

Component: symbolics

Keywords: gamma, incomplete, special, functions

Author: Ralf Stephan

Branch/Commit: 157b268

Reviewer: Buck Evan, Paul Masson

Issue created by migration from https://trac.sagemath.org/ticket/16697

The text was updated successfully, but these errors were encountered:

nbruin · 2014-07-22T15:25:54Z

comment:2

We don't need all of gamma, gamma_inc, incomplete_gamma as symbolic function if they all mean the same. Some of them are probably there for user convenience or because of legacy. I think we want only one spelling for gamma_upper, whatever it may be. The gamma(a,x) convention seems fairly prominent in computer algebra, but is a poor translation of math notation, where the difference between upper and lower is made with case (and indeed, the complete gamma function tends to be capital gamma. I don't think I've ever seen different).

Because of tab completion, you definitely want a binding available that starts with gamma. I'd be fine with gamma_lower but if you want to go with gamma_lower_incomplete, that's fine with me too. So, I'd propose

Maintain status quo with respect to upper incomplete gamma, i.e., don't change sage's behaviour for gamma(a,x) and gamma_inc(a,x). Just implement a symbolic function gamma_lower (or modified spelling) to correspond to maxima's gamma_greek.

rwst · 2014-07-22T16:02:07Z

comment:3

I don't underswtand. With "leave as is" you would keep gamma(a,x) as returned expression for the upper function? Or would you make gamma_inc fully symbolic in that it gets no longer converted to gamma(a,x)?

rwst · 2014-07-23T07:07:04Z

comment:4

The reason why gamma_inc(x,y) gets output as gamma(x,y) is not conversion as I thought but simple naming super in the constructor. It would be just as easy to make that gamma_inc and change a few doctests such that then:

         sage: f = exp_integral_e(-1,x)
         sage: f.integrate(x)
-        Ei(-x) - gamma(-1, x)
+        Ei(-x) - gamma_inc(-1, x)

             sage: gamma_inc(3,2)
-            gamma(3, 2)
+            gamma_inc(3, 2)

             sage: gamma(3,2)
             gamma_inc(3, 2)

             sage: gamma(x,0)
             gamma(x)

             sage: gamma(x, 0, hold=True)
-            gamma(x, 0)
+            gamma_inc(x, 0)

I'm appending the patch that enables this here, so it is preserved.

rwst · 2014-07-23T07:08:35Z

Attachment: 16697-gammainc-alt-patch.gz

see comment

rwst · 2014-07-23T08:10:44Z

Branch: u/rws/implement_symbolic_lower_incomplete_gamma_function

sagetrac-git · 2014-07-23T12:40:55Z

Commit: 93c2f94

sagetrac-git · 2014-07-23T12:40:55Z

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

`93c2f94`	`16697: deprecate "incomplete_gamma"`

sagetrac-git · 2014-07-23T14:43:23Z

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

`5c1dd67`	`16697: implement gamma_inc_lower`

sagetrac-git · 2014-07-23T14:43:23Z

Changed commit from 93c2f94 to 5c1dd67

sagetrac-git · 2014-07-26T07:02:49Z

Changed commit from 5c1dd67 to d8d7ee2

sagetrac-git · 2014-07-26T07:02:49Z

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

`d8d7ee2`	`16697: use mpmath as default evalf algorithm`

nbruin · 2014-08-08T14:58:01Z

comment:11

On sage-support Rafael Greenblatt reports

sage: (incomplete_gamma(x,y).diff(x)).simplify()
TypeError: ECL says: Error executing code in Maxima: gamma: wrong number of arguments.

The problem probably arises because the function ends up separated from its arguments:

sage: incomplete_gamma(x,y).diff(x)
D[0](gamma)(x, y)

which might confuse the translator (the behaviour is consistent with just going by strings). The changes here might improve the situation. It's worth checking.

The following of course works because it does NOT primarily look at strings:

sage: from sage.interfaces.maxima_lib import *
sage: maxima_calculus(sr_to_max(incomplete_gamma(x,y).diff(x)))
'diff(gamma_incomplete(x,y),x,1)

rwst · 2014-08-08T15:34:14Z

comment:12

Replying to @nbruin:

On sage-support Rafael Greenblatt reports
sage: (incomplete_gamma(x,y).diff(x)).simplify()
TypeError: ECL says: Error executing code in Maxima: gamma: wrong number of arguments.
The problem probably arises because the function ends up separated from its arguments:
sage: incomplete_gamma(x,y).diff(x)
D[0](gamma)(x, y)
which might confuse the translator (the behaviour is consistent with just going by strings). The changes here might improve the situation. It's worth checking.

Does the same with the patch. Maybe include the fix here? Is it simple?

nbruin · 2014-08-08T16:27:13Z

comment:13

Replying to @rwst:

Does the same with the patch. Maybe include the fix here? Is it simple?

Might not be too bad. I think the problem is in sage.symbolic.expression_conversions.InterfaceInit.derivative (line 515 or so). You can see there that the string representation of a derivative expression gets assembled by using f.name(). Replacing that by f._maxima_init_() might already do a better job:

sage: gamma(x,y).operator().name()
'gamma'
sage: gamma(x).operator().name()
'gamma'
sage: gamma(x,y).operator()._maxima_init_()
'gamma_incomplete'
sage: gamma(x).operator()._maxima_init_()
'gamma'

rwst · 2014-08-08T16:37:46Z

comment:14

That gives:

sage: (incomplete_gamma(x,y).diff(x)).simplify()
D[0](gamma)(x, y)

nbruin · 2014-08-08T18:43:12Z

comment:15

Replying to @rwst:

That gives:

sage: (incomplete_gamma(x,y).diff(x)).simplify()
D[0](gamma)(x, y)

Cool! so it IS easy. Be sure to also test (I haven't checked the answer is correct, but at least it shouldn't raise an error):

sage: (incomplete_gamma(x,x+1).diff(x)).simplify().simplify()
-(x + 1)^(x - 1)*e^(-x - 1) + D[0](gamma)(x, x + 1)

to check that both occurrences of f.name() have been replaced. I check the rest of that file and didn't find any further suspicious uses of name so after this, our maxima interface is probably perfect :-).

nbruin · 2014-08-08T22:48:25Z

comment:16

See #16785 for a more complete fix.

sagetrac-git · 2014-08-09T15:18:31Z

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

`862798b`	`Merge branch 'develop' into t/16697/implement_symbolic_lower_incomplete_gamma_function`
`705c34e`	`16697: fix a related bug`
`a73c5e3`	`16697: fix previous patch and a doctest`

sagetrac-git · 2014-08-09T15:18:31Z

Changed commit from d8d7ee2 to a73c5e3

rwst · 2014-08-09T15:20:45Z

comment:18

I also had to fix a doctest in integration where the given result didn't have enough precision because it came from maxima. Now the default is mpmath and that uncovered it.

sagetrac-git · 2014-08-10T09:28:10Z

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

`7558a7c`	`16697: revert fix in favor of merge of 16785`
`9c0f66d`	`trac #16785: improve differential operator translation to maxima`
`1fc19e7`	`Merge branch 't/16785/ticket/16785' into t/16697/implement_symbolic_lower_incomplete_gamma_function`

sagetrac-git · 2014-08-10T09:28:10Z

Changed commit from a73c5e3 to 1fc19e7

rwst · 2014-08-10T09:28:53Z

Merged: #16785

fredrik-johansson · 2016-02-19T10:10:06Z

comment:70

You could also compute the incomplete gamma function with Arb.

rwst · 2016-03-14T16:02:17Z

Changed branch from u/rws/16697-1 to u/rws/16697-2

rwst · 2016-03-14T16:03:07Z

Changed commit from 8661837 to c6be42f

rwst · 2016-03-14T16:03:07Z

Changed dependencies from sympy-0.7.7 to #20185

rwst · 2016-03-14T16:03:07Z

New commits:

`c6be42f`	`Merge branch 'u/rws/16697-1' of git://trac.sagemath.org/sage into i16697`

fchapoton · 2016-05-14T08:28:21Z

comment:73

does not apply

rwst · 2016-05-14T12:28:05Z

comment:74

Wanna review?

sagetrac-git · 2016-05-20T15:10:31Z

Changed commit from c6be42f to 157b268

sagetrac-git · 2016-05-20T15:10:31Z

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

`1f1c367`	`Merge branch 'develop' into t/16697/16697-2`
`157b268`	`remove some proposed but obsolete doctests`

paulmasson · 2016-06-26T22:01:45Z

comment:77

Current 7.3.beta5 Sage builds fine with these modifications. Doctests all pass. Documentation builds.

Random numeric tests accurate. Function plots. One curious difference from gamma_inc:

sage: gamma_inc_lower(a,x).diff(x)
x^(a - 1)*e^(-x)
sage: gamma_inc(a,x).diff(x)
D[1](gamma)(a, x)

Looks good to me. Sympy already updated: any other reason not to merge?

paulmasson · 2016-06-26T22:02:12Z

Changed reviewer from Buck Evan to Buck Evan, Paul Masson

rwst · 2016-06-27T07:16:18Z

comment:79

Thanks.

vbraun · 2016-06-27T16:22:10Z

Changed branch from u/rws/16697-2 to 157b268

rwst added this to the sage-6.3 milestone Jul 22, 2014

rwst added c: symbolics labels Jul 22, 2014

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vbraun closed this as completed in 7435287 Jun 27, 2016

eviatarbach mentioned this issue Jul 2, 2016

Implement symbolic confluent hypergeometric functions #14896

Closed

This was referenced Nov 26, 2014

incomplete gamma function bugs for certain arguments #17328

Closed

lower incomplete gamma as gamma(a,0,x) #17722

Open

rwst mentioned this issue Mar 17, 2015

add the gamma(a,0,x) interface for the lower gamma function #17899

Closed

This was referenced Jul 30, 2015

numerical bug in incomplete gamma function #18210

Closed

incomplete gamma identities #18956

Open

fchapoton mentioned this issue Oct 28, 2018

remove deprecated incomplete_gamma #26553

Closed

implement symbolic lower incomplete gamma function #16697

implement symbolic lower incomplete gamma function #16697

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