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Re: Root finding needs higher accuracy


> This problem is known to be difficult for solvers in double precision

Hi.  As a side note, look at the Newton version of the equation.

f[x_]:=x^2/(2 E^1)-(2 x)/Sqrt[E^1]+Log[x]+1

x-f[x]/f'[x]  //FullSimplify

(x (x^2-2 E Log[x])) / (2 (Sqrt[E]-x)^2)

Notice that as the next guess (x) approaches Sqrt[E], then the next guess approaches infinity, as the denominator goes to zero.

I think this explains why this is a hard problem, especially at machine precision.

So, my First guess would be Sqrt[E] using limits instead.

Limit[%,x->Sqrt[E]]

Sqrt[E]

Lucky guess!  ;>0    It happens to be the solution.

= = = = = = = = = = = = = =
HTH
Dana DeLouis
Mac, Ver 8
= = = = = = = = = = = = = =



On Nov 24, 7:04 am, HwB <hwborch... 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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