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Re: Root finding needs higher accuracy
*To*: mathgroup at smc.vnet.net
*Subject*: [mg123162] Re: Root finding needs higher accuracy
*From*: Bob Hanlon <hanlonr357 at gmail.com>
*Date*: Fri, 25 Nov 2011 04:54:30 -0500 (EST)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*References*: <201111241157.GAA29024@smc.vnet.net>
Tell Reduce or Solve that you are interested in a real root
f12[x_] := Log[x] + x^2/(2 Exp[1]) - 2 x/Sqrt[Exp[1]] + 1
Reduce[f12[x] == 0, x, Reals]
x == Root[{-4*Sqrt[E] + 4*#1 & ,
1.6487212707001281468486516320736540164304528843786101788282`20.\
311523332141462}]
Solve[f12[x] == 0, x, Reals] // Union
{{x -> Root[{-4*Sqrt[E] + 4*#1 & ,
1.6487212707001281468486516320736540164304528843786101788282`20.\
311523332141462}]\
}}
Or restrict x to be positive
Reduce[{f12[x] == 0, x > 0}, x]
x == Root[{-4*Sqrt[E] + 4*#1 & ,
1.6487212707001281468486516320736540164304528843786101788282`20.\
311523332141462}]
Solve[{f12[x] == 0, x > 0}, x] // Union
{{x -> Root[{-4*Sqrt[E] + 4*#1 & ,
1.6487212707001281468486516320736540164304528843786101788282`20.\
311523332141462}]\
}}
It is not clear to me why Mathematica doesn't automatically complete the solve
Solve[-4 Sqrt[E] + 4 x == 0, x] // Flatten
{x -> Sqrt[E]}
Log[x] + x^2/(2 Exp[1]) - 2 x/Sqrt[Exp[1]] + 1 == 0 /. %
True
Bob Hanlon
On Thu, Nov 24, 2011 at 6:57 AM, HwB <hwborchers at googlemail.com> wrote:
> I would like to numerically find the root of the following function
> with up to 20 digits.
>
> f12[x_] := Log[x] + x^2 / (2 Exp[1]) - 2 x / Sqrt[Exp[1]] + 1
>
> This problem is known to be difficult for solvers in double precision
> arithmetics. I thought it should be easy with Mathematica, but the
> following attempts were not successful.
>
> SetPrecision[
> x /. FindRoot[f12[x], {x, 1.0, 3.4}, Method -> "Brent",
> AccuracyGoal -> Infinity, PrecisionGoal -> 20], 16]
> # 1.648732212532746
> SetPrecision[
> x /. FindRoot[f12[x], {x, 1.0, 3.4}, Method -> "Secant",
> AccuracyGoal -> Infinity, PrecisionGoal -> 20], 16]
> # 1.648710202030051
>
> The true root obviously is Sqrt[Exp[1]]//N = 1.648721270700128...
>
> The symbolic solver explicitely says it cannot solve this expression.
> What do I need to do to get a much more exact result out of
> Mathematica?
>
> Many thanks, Hans Werner
>
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