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Re: Change in NSolve algorithm

  • To: mathgroup at smc.vnet.net
  • Subject: [mg98560] Re: Change in NSolve algorithm
  • From: Szabolcs Horvát <szhorvat at gmail.com>
  • Date: Mon, 13 Apr 2009 03:30:59 -0400 (EDT)
  • References: <grpif7$lnr$1@smc.vnet.net> <grs6as$qie$1@smc.vnet.net>

ADL wrote:
> What surprised me about Wilkinson's polynomial example, which should
> return Range[20], is the following.
> 
> Mathematica Help says that: "NSolve[eqns,vars,n] gives results to n-
> digit precision". Consequently, omitting n should imply machine
> precision. Nevertheless:
> 
> sols = x /. NSolve[Expand[Product[x - i, {i, 1, 20}] == 0, x], x]
> 
> {1.,2.00000000000096,2.99999999986639,4.0000000049587,4.99999990255529,6.00=
> 000083841557+1.13297570609149*10^-9
> I,7.00008136819274-0.0000108589979212047 I,
> 8.00087475000192+0.000151959812492809 I,
> 9.0002388679906+0.000147736991612302 I,
> 10.000992367067-0.000579336748622785 I,
> 11.0030045370741-0.00438286449628658 I,
> 12.0383486261734-0.102068022121597 I,
> 13.1275500318654+0.0361732078353353 I,
> 13.9803341855889+0.0168546889387958 I,
> 14.9015299203113-0.0418410427904214 I,
> 15.8215727528799+0.0440619301323384 I,
> 16.9944217046715-0.00669060159139528 I,
> 18.0000979553809-0.000170510501593321 I,
> 18.999972121053+0.00014011316501936 I,
> 19.9999979459428+1.38662076005353*10^-6 I}
> 
> while
> 
> sols = x /. NSolve[Expand[Product[x - i, {i, 1, 20}] == 0, x], x,
> $MachinePrecision]
> {1.000000000000000,2.000000000000000,3.000000000000000,4.000000000000000,5.=
> 000000000000000,6.000000000000000,7.000000000000000,8.000000000000000,9.000=
> 000000000000,10.00000000000000,11.00000000000000,12.00000000000000,13.00000=
> 000000000,14.00000000000000,15.00000000000000,16.00000000000000,17.00000000=
> 000000,18.00000000000000,19.00000000000000,20.00000000000000}
> 
> Even with 2 digits precision the result is correct!
> 
> sols = x /. NSolve[Expand[Product[x - i, {i, 1, 20}] == 0, x], x, 2]
> {1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.,11.,12.,13.,14.,15.,16.,17.,18.,19=
> .,20.}
> 
> So, I would say that your example indicates a misbehavior or a bug in
> NSolve.
> 
> Any comments?

Yes,  WorkingPrecision -> $MachinePrecision will use Mathematica's 
arbitrary precision arithmetic with $MachinePrecision number of digits 
(around 16 on most machines).  WorkingPrecision -> MachinePrecision will 
actually use the FPU to do the calculations.


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