Root finding needs higher accuracy

*To*: mathgroup at smc.vnet.net*Subject*: [mg123135] Root finding needs higher accuracy*From*: HwB <hwborchers at googlemail.com>*Date*: Thu, 24 Nov 2011 06:57:06 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

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

**Follow-Ups**:**Re: Root finding needs higher accuracy***From:*DrMajorBob <btreat1@austin.rr.com>

**Re: Root finding needs higher accuracy***From:*DrMajorBob <btreat1@austin.rr.com>

**Re: Root finding needs higher accuracy***From:*Bob Hanlon <hanlonr357@gmail.com>

**Re: Root finding needs higher accuracy***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**Re: Root finding needs higher accuracy***From:*"Harvey P. Dale" <hpd1@nyu.edu>