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

• trac_9130_combined.2.patch​

[fredrik.johansson]

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]

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]

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]

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]

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.

[ktkohl]

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]

fix segfault in py_float

[burcin]

Hi Karen,

sorry for taking so long to look at this. It seems that I forgot to check for a NULL pointer in py_float(). attachment:trac_9130-py_float_segfault.patch​ fixes the segfault.

sage: from sage.functions.other import beta
sage: beta(3,2.0)
0.0833333333333333

Will you have time to finish the patch?

You need to add an import statement to sage/functions/all.py so beta is available at the command line. The documentation also needs some care. IIRC there should be an empty line after INPUT, OUTPUT and EXAMPLES. The statement

        It is computed by various libraries within Sage, depending on
        the input type.

is too vague. We should either remove it or explain how GiNaC evaluates this (see beta_eval() and beta_evalf() in ​inifcns_gamma.cpp).

[ktkohl]

include beta in random_tests

[ktkohl]

Apply all patches in the order they were added:

• trac_9130_beta_function.patch

• trac_9130-py_float_segfault.patch

• trac_9130_beta_function.2.patch

• trac_9130_beta_function.3.patch

[benjaminfjones]

I think we discovered that the only complex inputs that the code accepts are ones where one of the parameters is equal to 1. In that case beta(1,x) = 1/x is used to compute the result. Looks like GiNaC can't handle complex inputs at all (or perhaps complex numbers aren't being passed to GiNaC in a way it understands).

On the other hand, ​mpmath does support evaluation at arbitrary precision complex numbers so that could be a useful enhancement that could take place in a new ticket.

I would change the docstrings to clearly indicate that

1. only real inputs are accepted (for now)
2. beta(1,x) = beta(x,1) = 1/x simplification is automatically applied

[ktkohl]

updated comments/examples/error message text.

[benjaminfjones]

combined reviewer patch folding up previously uploaded 5 patches

[benjaminfjones]

I uploaded a combined *single* patch that folds together the previously uploaded 5 patches applied in order. The patch looks great to me. I have one question about what happens when GinacFunction's __call__ returns a TypeError whose message doesn't contain "cannot coerce". What gets returned in that case? (the comments in the gamma implementation that the try / except block was borrowed from say that TypeErrors? are raised when a fast float is passed. These are then ignored in the current implementation? I haven't tested this.

I'm running make ptestlong on Sage-4.8.alpha6 with the combined patch applied overnight; will update in the morning.

[burcin]

I uploaded a new patch attachment:trac_9130-py_float_segfault.take2.patch​ which checks for TypeError instead of ValueError in sage.symbolic.pynac.py_float(). Now we can evaluate beta() on complex values, so the __call__() method in Karen's patches can be removed.

[ktkohl]

new combined file replacement by trac_9130-py_float_segfault.take2.patch and removal of __call__() method

[ktkohl]

New patch attachment:trac_9130_combined.2.patch​ replaces burcin's old patch with the newer one and also removes the __call__() method in beta.

Apply only this one.