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Re: Simplyfing Sum[Mod[f(k),y],{k,k0,n}] type expressions? (Newbie question)
*To*: mathgroup at smc.vnet.net
*Subject*: [mg40219] Re: [mg40202] Simplyfing Sum[Mod[f(k),y],{k,k0,n}] type expressions? (Newbie question)
*From*: Daniel Lichtblau <danl at wolfram.com>
*Date*: Wed, 26 Mar 2003 02:42:34 -0500 (EST)
*References*: <200303251950.OAA11794@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
prodogoss wrote:
>
> I'm dealing with a problem that requires lots of summations of (x mod
> y) type expressions. I had hoped that Mathematica (which I've just
> started using) could simplify these but it doesn't seem to work.
>
> E.g., why won't (can't?) Mathematica simplify the expression:
>
> Sum[Mod[3n-k-4, 3],{k, 2n-1,d}] ?
>
> By *hand* I can show that if
>
> S = SUM((3n-k-4) mod 3, k = 2n-1 to d) and
>
> G = d - 2n + 2, then S = f(t) where t = (n - 2 - G) mod 3, and
>
> (f(0), f(1), f(2)) = ({1, 0, 0}, {0, 1, 0}, {0, 0, 1})(G-1, G, G)
>
> ie 3x1 matrix = (3rd order unit matix) x (3x1 matrix)
>
> Hence, given values of d & n, S can be expressed in such a way that
> its calculation is possible without having to labourously evaluate S
> for each step, k.
>
> I had heard so much about Mathematica and thought it would excel at
> handling problems like this involving summations of Mod[] etc.? Is
> there another way of doing this? Any help would be great!
>
> Thanks in advance!
In[1]:= InputForm[PolynomialMod[Sum[3*n-k-4, {k,2*n-1,d}], 3]]
Out[1]//InputForm= 2 + d^2 + 2*n + 2*n^2
Alternatively you can do it as you originally intended, but use
PolynomialMod rather than Mod as the latter expects NumberQ arguments.
In[2]:= InputForm[Sum[PolynomialMod[3*n-k-4,3], {k,2*n-1,d}]]
Out[2]//InputForm= 2 + 3*d + d^2 + 2*n - 4*n^2
Note that the results are equivalent via PolynomialMod[...,3].
Daniel Lichtblau
Wolfram Research
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