Services & ResourcesWolfram ForumsMathGroup Archive

[Date Index] [Thread Index] [Author Index]

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>