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Re: Re: Change in NSolve algorithm

  • To: mathgroup at smc.vnet.net
  • Subject: [mg98593] Re: [mg98545] Re: Change in NSolve algorithm
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 13 Apr 2009 03:37:01 -0400 (EDT)
  • References: <grpif7$lnr$1@smc.vnet.net> <200904120747.DAA27216@smc.vnet.net>

On 12 Apr 2009, at 16:47, 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. $MachinePrecision is not the same as MachinePrecision. Try the  
latter and see what happens.

Andrzej Kozlowski



>
>
> Alberto
>
>
> On 11 Apr, 09:56, mark mcclure <mcmcc... at unca.edu> wrote:
>> Has anyone noticed that the algorithm used by NSolve has
>> changed in V7?
>> ...
>> On the other hand, V7 fares slightly worse with Wilkinson's
>> polynomial:
>>  sols = x/.NSolve[Expand[Product[x-i, {i,1,20}] == 0,x]];
>>  Norm[Table[sols[[i]] - i, {i, 1, 20}]]
>> ...
>> Mark McClure
>
>



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