Re: How to eliminate noises? A better way perhaps.

*To*: mathgroup at smc.vnet.net*Subject*: [mg122673] Re: How to eliminate noises? A better way perhaps.*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Sat, 5 Nov 2011 04:48:00 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201111021121.GAA03503@smc.vnet.net> <j8tl5t$f3t$1@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

Very nice, Richard. We shouldn't be so quick to get the hammer if we think we've seen a nail! Bobby On Fri, 04 Nov 2011 05:59:20 -0500, Richard Fateman <fateman at cs.berkeley.edu> wrote: > On 11/3/2011 1:59 AM, Bob Hanlon wrote: >> Use higher precision. > .. so say you all... > Well, that's the brute force method, and one reason to recommend it is > it doesn't require much thought. > > On the other hand, the person asking the question has an apparently > large expression which he knows has a double zero at n=1/2 and he wants > to know the behavior of the expression from 0.35 to 0.53. Namely, > around that zero. > > Let us call that expression p. A non-brute force, but mostly automatic > method is to note that (since p is a polynomial of degree 29) it is > EXACTLY equal to s, where > > s = Normal[Series[p, {n, 1/2, 29}]] > > A plot of s can be done in ordinary float arithmetic and looks to the > eye just like the plot of p, done with high working precision. Unique to > computer algebra systems, it is also possible to look at the structure > of s, and note that expanded around n=1/2 it is mighty sparse. > > Let k = n-1/2 . The expression being plotted is then > > 14 k^2 - 1296 k^5 + 1451188224 k^11 - 2002639749120 k^14 + > 598075300380672 k^17 - 83414577743659008 k^20 + > 3977900382788517888 k^23 - 113721152119718805504 k^26 + > 1516282028262917406720 k^29 > > > Sometimes a better way to format expressions for evaluation is to use > HornerForm. In this case it is > > > k^2 (14 + > k^3 (-1296 + .... > > Evaluating this expression at a value for k requires computing > k^2,k^3,k^6,and some other arithmetic for a total of about 18 arithmetic > operations, which can be done in double-float. > The real win here is that you can show how much more insight you might > get from using computer algebra. > > RJF > -- DrMajorBob at yahoo.com

**References**:**How to eliminate noises?***From:*Artur <grafix@csl.pl>