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Re: Root finding needs higher accuracy

• To: mathgroup at smc.vnet.net
• Subject: [mg123171] Re: Root finding needs higher accuracy
• From: "Harvey P. Dale" <hpd1 at nyu.edu>
• Date: Fri, 25 Nov 2011 04:56:08 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201111241157.GAA29024@smc.vnet.net>

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: [mg123171] 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

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