Higher precision in Error function Erf[] needed.

• To: mathgroup at smc.vnet.net
• Subject: [mg127038] Higher precision in Error function Erf[] needed.
• From: Cyril <cyril.stephanos at gmail.com>
• Date: Tue, 26 Jun 2012 04:48:42 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```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

```

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