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Topic: Approximations
Replies: 23   Last Post: Sep 3, 2000 2:01 PM

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Joe Ascoly

Posts: 23
Registered: 12/13/04
Re: Approximations
Posted: Sep 3, 2000 2:01 PM
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More suitable than previously suggested scheme is Gerlach's scheme (possible
extension of the classic Newton method) for square roots. The convergence is
"decadic" , meaning that the number of good digits expands tenfold on each
iteration

Hi,>
> Does anyone know of any approximations to the square root function thatare
> more computationaly efficient. (I need to it for implementation in>

hardware).
>> thanks for any help
-------------------------------------
A method other than Newton iteration was used in the Fortran library for
the square root.
This describes how the square root was done
in software for a system that had three
precisions of floating point arithmetic,
called short, long and extended with 4,8,16 bytes,
using base 16. The extended precision arithmetic did
not include a divide which had to be simulated.
Here is an outline of how the square root was done.
There were 2 iterations in short precision, one in
long and one in extended arithemetic.
Scaling the input argument is used to avoid possible
intermediate underflows. The first approximation y
is computed as:
y0= 16^16(1.807018 - 1.576942/(m+.950356)
where m is the mantissa of the argument between 1/16 and 1.
and the maximum relative error of y0 is 2^-5.48
The final interation carried out in extended precision:
y4=y3 - 2y3*((y3^2-x)/(3y3^2+x))
This is an equivalent form due to Richard Dedekind
in 1830's and published in Survey of Numerical Analysis
J Todd 1962
In the process of combining terms a rounding bias is
introduced to attain results which are almost always
properly rounded. For the vector machines the square
root was improved to always round properly due to a
method devised by Bryant Tuckerman.
See System Journal article FORTRAN extended precision
library by the late H Kuki and J Ascoly V10N1 1971.
The square root of a negative number should result in
an error message. Hardware handles exceptions
by using interupts.








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