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Re: need a quicker method for SquaresR func

• To: mathgroup at smc.vnet.net
• Subject: [mg121231] Re: need a quicker method for SquaresR func
• From: Dana DeLouis <dana01 at me.com>
• Date: Mon, 5 Sep 2011 07:06:52 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

> SquaresR ... like  JacobiTheta  OR something else....

Hi.  I see you don't have an answer.
I don't either, but I thought this was interesting

http://mathworld.wolfram.com/SumofSquaresFunction.html

Item # 39 had a code for R4.

Downloading the workbook gave this as 1 alternative as a generating 
function, but I'm afraid it's slower.

fx[n_]:=Coefficient[Series[EllipticTheta[3,0,q]^4,{q,0,n}],q,n]

fx[10000]//Timing
{4.65, 18744}

SquaresR[4,10000]//Timing
{0.00008, 18744}

This version was faster, but not as fast as the built-in function:

8 Plus@@Select[Divisors[100000000],Mod[#,4]!=0&]//Timing
{0.002715,11718744}

SquaresR[4,100000000]//Timing
{0.000972,11718744}

As a side note, it looks like the solution is about 2-3 times the size 
of the input, on average.
So, I would guess your answer is larger than your input.
Sounds like a tough problem with inputs that large.

t=Table[SquaresR[4,n]/n //N, {n,1000,300000,1000}]  ;

{Mean[t],StandardDeviation[t]}
{2.956801,1.680969}

= = = = = = = = = =
Good Luck.  :>)
Dana DeLouis
$Version
8.0 for Mac OS X x86 (64-bit) (November 6, 2010)





On Aug 25, 7:12 am, Time Traveler <time.traveler... at gmail.com> wrote:
> Hi;
> For a research project I need a quicker way to obtain SquaresR
> function.  example --> SquaresR[4 ,  H] . The problem is ;
>
> Integer H is quite large (about 200 digit)  and it takes exceedingly
> long time to return the ans.  Is there a quicker way to find this
> stated function by a different methods, like  JacobiTheta  OR
> something else....
>
> Thanks for help in advance
> Time