Re: Root finding needs higher accuracy
- To: mathgroup at smc.vnet.net
- Subject: [mg123177] Re: Root finding needs higher accuracy
- From: Peter Falloon <pfalloon at gmail.com>
- Date: Fri, 25 Nov 2011 04:57:13 -0500 (EST)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <jalbrn$sib$1@smc.vnet.net>
On Nov 24, 11:04 pm, HwB <hwborch... 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, PrecisionGoa=
l -> 20], 16]
> # 1.648732212532746
> SetPrecision[
> x /. FindRoot[f12[x], {x, 1.0, 3.4}, Method -> "Secant",
> AccuracyGoal -> Infinity, PrecisionGoa=
l -> 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
You need to increase the WorkingPrecision:
In[658]:= f[x_] := Log[x] + x^2 / (2 Exp[1]) - 2 x / Sqrt[Exp[1]] + 1
x0 = x /. FindRoot[f[x], {x, 1.5}, PrecisionGoal->20, WorkingPrecision-
>30];
{x0, f[x0]}
Out[660]= {1.64872127035488735592446035944, 0.*10^-30}
Cheers,
Peter.
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