Re: Multinomial coefficients evaluation
- To: mathgroup at smc.vnet.net
- Subject: [mg68138] Re: Multinomial coefficients evaluation
- From: "sashap" <pavlyk at gmail.com>
- Date: Tue, 25 Jul 2006 04:01:45 -0400 (EDT)
- References: <ea1nis$pkr$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Hi Michal,
You would be much better off posting here input forms of the input, than
the whole notebook. It has a benefit of readability without having access
to Mathematica, and does not clutter the forum.
Since
In[3]:=
Attributes[Sum]
Out[3]=
{HoldAll,Protected,ReadProtected}
It evaluates its arguments later in the game, and for every term in the
sum, resuling
in multitude of subtitutions, when all you need is one substitution
done before the summing commences. Thus, wrapping Evaluate around the
first argument helps
to speed it up.
In[1]:=
Timing[s1 = Sum[Evaluate[Multinomial[n1, n2, n3]*Subscript[a, 1]^n1*
Subscript[a, 2]^n2*Subscript[a, 3]^n3 /. n3 -> n - n1 - n2],
{n1, 0, n},
{n2, 0, n - n1}]; ]
Out[1]=
{0.17200000000000004*Second, Null}
In[2]:=
Timing[s2 = Sum[Evaluate[((n1 + n2 + n3)!/(n1!*n2!*n3!))*Subscript[a,
1]^n1*
Subscript[a, 2]^n2*Subscript[a, 3]^n3 /. n3 -> n - n1 - n2],
{n1, 0, n},
{n2, 0, n - n1}]; ]
Out[2]=
{0.07799999999999985*Second, Null}
The reason for slagishness of Multinomial is in its Orderless
attribute:
In[21]:= Attributes[Multinomial]
Out[21]= {Listable, NumericFunction, Orderless, Protected,
ReadProtected}
Multinomial sorts its arguments for every term evaluation.
Oleksandr Pavlyk
Wolfram Research
Michal Kvasnicka wrote:
> Why is function Multinomial so slower than its direct definition??? See
> attached notebook with three equivalent ways of the simple multinomial
> series expansion.
>
> Michal Kvasnicka
>
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