Access to beta function

[kcrisman]

Although Maxima has the beta function, Sage doesn't:

sage: a, b, c = var('a b c') 
sage: assume(a > 0) 
sage: assume(b > 0) 
sage: x = var('x') 
sage: beta_dist = x**(a-1) * (1 - x)**(b-1) 
sage: c = integral(beta_dist, x, 0, 1) 
sage: c
beta(a, b)
sage: c(a=.5,b=.5)
beta(0.500000000000000, 0.500000000000000)
sage: c(a=.5,b=.5).n()
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)

/Users/karl-dietercrisman/<ipython console> in <module>()

/Users/karl-dietercrisman/Desktop/sage-4.4.2/local/lib/python2.6/site-packages/sage/symbolic/expression.so in sage.symbolic.expression.Expression.n (sage/symbolic/expression.cpp:17042)()

TypeError: cannot evaluate symbolic expression numerically

This is is Ginac, though, and there is even room for defining it in symbolic/expression.pyx . It probably is also included in some of our other libraries, as a standard special function.

---

Apply

• Patches at Implement a symbolic version of the arg function #4498 and Mark random symbolic expression doctests with #random #12507
• attachment: trac_9130-beta_function.3.patch
• attachment: trac_9130-py_float_segfault.take2.patch
• attachment: trac_9130-reviewer.patch
• attachment: trac_9130-random-tests.2.patch

Depends on #4498
Depends on #12507

CC: @benjaminfjones

Component: symbolics

Keywords: special function, pynac, sd35.5 Cernay2012

Author: Karen Kohl, Burcin Erocal, Karl-Dieter Crisman

Reviewer: Benjamin Jones, Burcin Erocal, Karl-Dieter Crisman

Merged: sage-5.0.beta6

Issue created by migration from https://trac.sagemath.org/ticket/9130

[fredrik-johansson]

comment:1

For numerical evaluation, mpmath has beta and also the generalized incomplete beta function for complex arguments. But it's probably easy to do the complete beta function directly with Ginac.

Simplification for rational arguments (beta(0.5,0.5) = pi) would be nice.

Unless someone else wants to work on this, I might have a stab at it within a couple of days.

[kcrisman]

comment:2

I wasn't suggesting which of the several packages in Sage should be used for numerical evaluation, though mpmath did come to mind :-)

I don't think that beta(0.5,0.5) would work, given that

sage: gamma(0.5)
1.77245385090552
sage: gamma(1/2)
sqrt(pi)

but beta(1/2,1/2) becoming pi should work fine once we have a symbolic wrapper (with or without Ginac):

sage: maxima_console()
Maxima 5.21.1 http://maxima.sourceforge.net
using Lisp ECL 10.4.1
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) beta(1/2,1/2);
(%o1)                                 %pi

Please do try to add this! We definitely often get email asking for various special functions both symbolically and numerically. Also, the more examples we have, the easier it is to finish off the rest of them by cut and paste.

[burcin]

comment:3

GiNaC has a beta function, so this can probably be solved simply by wrapping that. See #8864 for an example.

Though I don't know why the beta() function in sage/symbolic/expression.pyx is commented. Maybe there is something I'm missing.

Fredrik, it would be great if you can do this. I'd be happy to answer questions if anything goes wrong.

[kcrisman]

comment:4

Replying to @burcin:

GiNaC has a beta function, so this can probably be solved simply by wrapping that. See #8864 for an example.

Though I don't know why the beta() function in sage/symbolic/expression.pyx is commented. Maybe there is something I'm missing.

I think the same reason the psi and psi2 ones are commented - when those were implemented, they didn't notice that they had been commented earlier. This was probably pretty early in the conversion, maybe when William was dealing with CLN (whatever that is)?

[burcin]

comment:5

Replying to @kcrisman:

Replying to @burcin:

GiNaC has a beta function, so this can probably be solved simply by wrapping that. See #8864 for an example.

Though I don't know why the beta() function in sage/symbolic/expression.pyx is commented. Maybe there is something I'm missing.

I think the same reason the psi and psi2 ones are commented - when those were implemented, they didn't notice that they had been commented earlier. This was probably pretty early in the conversion, maybe when William was dealing with CLN (whatever that is)?

psi() and psi2() were commented because at the time there was no method defined to numerically evaluate those. This is not the case for beta() however. Here is the evalf method (from line 227 of ginac/inifcns_gamma.cpp):

	if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y)) {
		try {
			return exp(lgamma(ex_to<numeric>(x))+lgamma(ex_to<numeric>(y))-lgamma(ex_to<numeric>(x+y)));
		} catch (const dunno &e) { }
	}

We'll find out when someone tries this out I suppose.

[sagetrac-ktkohl]

Attachment: trac_9130_beta_function.patch.gz

[sagetrac-ktkohl]

comment:7

Added ginac wrapper for beta function.
There is a problem when one argument is a float.

--Karen

sage: beta(3,2.0)


------------------------------------------------------------
Unhandled SIGSEGV: A segmentation fault occurred in Sage.
This probably occurred because a *compiled* component
of Sage has a bug in it (typically accessing invalid memory)
or is not properly wrapped with _sig_on, _sig_off.
You might want to run Sage under gdb with 'sage -gdb' to debug this.
Sage will now terminate (sorry).
------------------------------------------------------------

[burcin]

Attachment: trac_9130-py_float_segfault.patch.gz

fix segfault in py_float