EXPLANATION FILE OF PROGRAM ASYMERF
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The Error Function
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The error function erf(x) can be approximated with an asymptotic series.
x -t^2 dt
erf(x) = 2/sqrt(pi) Sum e
0
This function or its complement, erfc(x) = 1 - erf(x), is often involved
in the solutions to heat transfer and ionic diffusion problems. It also
appears in statistics. In many cases, there is need to know the value of
erf(x) for large values of the argument ( x > 3); tables seldom cover
this range adequately.
Peirce gives a convenient asymptotic expansion for positive values of x:
1 1 x 3 1 x 3 x 5
erf(x) = 1 - ---- + -------- - --------- + ...
2x^2 (2x^2)^2 (2x^2)^3
As with most asymptotic series, the above expression eventually diverges.
However, we can again terminate the summation at the smallest term and
use the magnitude of the next term as a measure of the error bound. This
is done in program ASYMERF: The input is X, the returned values are
Y = erf(X), Y1 = erfc(X), the error estimate E and the number of terms
summed, N.
For x <= 3, a rapidly convergent Taylor series approximation can be used:
3 5 7
erf(x) = (2/sqrt(pi)) (x - x / 3 + x / (5 * 2!) - x / (7*3!) + ...)
Extract from Reference [BIBLI 01].
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End of file asymerf.txt