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 >

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