Re: Higher precision in Error function Erf[] needed.

• To: mathgroup at smc.vnet.net
• Subject: [mg127098] Re: Higher precision in Error function Erf[] needed.
• From: Ray Koopman <koopman at sfu.ca>
• Date: Sat, 30 Jun 2012 05:16:55 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jsbt4i\$6pq\$1@smc.vnet.net> <jseeua\$i3d\$1@smc.vnet.net> <jsjqhg\$d2j\$1@smc.vnet.net>

```On Jun 29, 1:55 am, Cyril <cyril.stepha... at gmail.com> wrote:
> On Wednesday, June 27, 2012 10:06:34 AM UTC+2, Ray Koopman wrote:
>> On Jun 26, 1:50 am, Cyril <cyril.stepha... at gmail.com> wrote:
>>> Hi everyone
>>>
>>> I am using the error function as part of differential equations I
>>> am solving with NDSolve. Unfortunately, for certain parameters, the
>>> error function Erf[] starts oscillating and yielding wrong values.
>>>
>>> The oscillation does not show when I plot the error function itself,
>>> but when I plot this expression
>>>
>>> Plot[Exp[8*(10 - Abs[10 - 20000*x])]*(1 - Erf[(8*(10 - Abs[10 -
>>> 20000*x]))^(1/2)]), {x, 0, 1*10^(-3)}, PlotRange -> Full]
>>>
>>> it becomes visible.
>>>
>>> When I do the integration for error function with NDSolve manually
>>> and plot the expression, the oscillation vanishes for high values
>>> of PrecisionGoal and WorkingPrecision, e.g.
>>>
>>> Plot[ Exp[8*(10 - Abs[10 - 20000*x])]*(1 - NIntegrate[ 2/Sqrt[Pi]
>>> Exp[-t^2], {t, 0, (8*(10 - Abs[10 - 20000*x]))^(1/2)},
>>> PrecisionGoal -> 40, WorkingPrecision -> 40]), {x, 0, 1*10^(-3)},
>>> PlotRange -> Full]
>>>
>>> The oscillation of Erf[] is not only visible in the plot,
>>> it also leads to wrong results in my calculations.
>>>
>>> I tried to increase the precision of Erf[] by using N[Erf[],n]
>>> with very high values for n - as described in the Mathematica help
>>> - but it does not change anything.
>>>
>>> Solving the Integral for the error function manually with NDSolve
>>> yields two problems. It is very slow - it takes about three minutes
>>> on my computer (I have to solve many equations) - whereas Erf[] is
>>> much faster. And it seems that I cannot use NDSolve as part of an
>>> equation I want to solve with NDSolve itself.
>>>
>>> Is there a possibility to increase the precision of Erf[]
>>> and obtain more accurate results?
>>>
>>>
>>> Best regards
>>> Cyril
>>
>> Change  (1 - Erf[...])  to  Erfc[...] .
>
> Ok, that was easy and works perfect. Thanks a lot!
>
> Although it was not what I expected.
> Is there a possibility to alter the precision of Erf[]?
>
> Best regards
> Cyril

Just alter the precision of its input. First define

f[x_,n_] := 8*(10 - Abs[10 - 20000*SetPrecision[x,n]])

Then

Plot[Exp@#*(1-Erf@Sqrt@#)&@f[x,20], {x,0,10^-3}, Compiled->False]

works fine.

```

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