Re: Root finding needs higher accuracy

*To*: mathgroup at smc.vnet.net*Subject*: [mg123172] Re: Root finding needs higher accuracy*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Fri, 25 Nov 2011 04:56:19 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201111241157.GAA29024@smc.vnet.net>

This problem is trivial for Mathematica, because Mathemtica can solve it *exactly*: Reduce[ Log[x] + x^2/(2 Exp[1]) - 2 x/Sqrt[Exp[1]] + 1 == 0, x, Reals] x==Root[{4 #1-4 Sqrt[E]&,1.64872127070012814685}] N[%, 100] x==1.648721270700128146848650787814163571653776100710148011575079311640661021194215608632776520056366643 Andrzej Kozlowski On 24 Nov 2011, at 12:57, HwB 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 >

**References**:**Root finding needs higher accuracy***From:*HwB <hwborchers@googlemail.com>