[ATM] Raytracing Function

["Richard F.L.R. Snashall" <rflrs@rcn.com>]

One function that keeps recurring in [algebraic] raytracing is
what I have come to call an offset root (or offset square root):

    OffsetRt( a, x ) = sqrt( a^2 + x ) - a

I am trying to implement this as a Java function with the
following basic algorithm:

    root = sqrt( |a| * |a| + x ) - |a|

    root = 0.5 * ( root * root + x ) / ( |a| + root )
    root = 0.5 * ( root * root + x ) / ( |a| + root )
                   ...

    root = root + ( |a| - a )

The first line creates an initial guess using the base square root
function.  The middle line[s] apply Newton-Raephson iteration to
refine the root.  The last line adjusts the solution for negative
values of parameter a.

When |x| is small relative to |a|, square root in the initial guess
and the final term nearly cancel; the result can have a large amount
of relative error.  However, in that case the iteration converges
quickly.  I have reason to suspect, then, that the number of
Newton-Raephson iterations needed is quite low, maybe even just one.

Opinions?

    
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