Re: Root finding needs higher accuracy

*To*: mathgroup at smc.vnet.net*Subject*: [mg123208] Re: Root finding needs higher accuracy*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Sat, 26 Nov 2011 05:09:20 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201111241157.GAA29024@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

Harvey, Sadly, that result is correct to just 16 decimal places: root = x /. FindRoot[Log[x] + x^2/(2 Exp[1]) - 2 x/Sqrt[Exp[1]] + 1 == 0, {x, 1}, WorkingPrecision -> 50] Replace[RealDigits@N[root, 25] - RealDigits@N[Sqrt@E, 25] // First, {x : Longest[0 ..], ___} :> Length@{x}] 1.6487212707001280730589817271865607686229236280618 16 Increasing WorkingPrecision to 80 throws FindRoot::cvmit and gets us two decimals closer: root = x /. FindRoot[Log[x] + x^2/(2 Exp[1]) - 2 x/Sqrt[Exp[1]] + 1 == 0, {x, 1}, WorkingPrecision -> 80] Replace[RealDigits@N[root, 25] - RealDigits@N[Sqrt@E, 25] // First, {x : Longest[0 ..], ___} :> Length@{x}] 1.64872127070012814492925334035434741717247563420986813745794650261295\ 21442685296 18 Bobby On Fri, 25 Nov 2011 03:56:08 -0600, Harvey P. Dale <hpd1 at nyu.edu> wrote: > Hans: > > The option you need is WorkingPrecision. > > FindRoot[Log[x]+x^2/(2 Exp[1])-2 > x/Sqrt[Exp[1]]+1==0,{x,1},WorkingPrecision->50] > > Best, > > Harvey > > > -----Original Message----- > From: HwB [mailto:hwborchers at googlemail.com] > Sent: Thursday, November 24, 2011 6:57 AM > To: mathgroup at smc.vnet.net > Subject: Root finding needs higher accuracy > > 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 > > -- DrMajorBob at yahoo.com

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