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Re: Root finding needs higher accuracy
*To*: mathgroup at smc.vnet.net
*Subject*: [mg123221] Re: Root finding needs higher accuracy
*From*: Dana DeLouis <dana01 at me.com>
*Date*: Sun, 27 Nov 2011 04:13:24 -0500 (EST)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
> 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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