Re: Change in NSolve algorithm

• To: mathgroup at smc.vnet.net
• Subject: [mg98545] Re: Change in NSolve algorithm
• From: ADL <alberto.dilullo at tiscali.it>
• Date: Sun, 12 Apr 2009 03:47:48 -0400 (EDT)
• References: <grpif7\$lnr\$1@smc.vnet.net>

```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.

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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