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

• To: mathgroup at smc.vnet.net
• Subject: [mg127093] Re: Higher precision in Error function Erf[] needed.
• From: Cyril <cyril.stephanos at gmail.com>
• Date: Fri, 29 Jun 2012 04:52:59 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jsbt4i\$6pq\$1@smc.vnet.net> <jseeua\$i3d\$1@smc.vnet.net>

```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

```

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