Absolute Value and Square Root implementations in fixed point

If by "absolute" you mean "magnitude" of a signed value then I explained that in my first reply.

There are only two meanings for "absolute" either the magnitude of a signed value or the magnitude of a complex pair. Both have been explained.

So I assume you are dealing with real signal (no imaginary component), hence all you have to do is get magnitude of negative value by inverting mag bits then add 1. Then use it to address LUT and then multiply back by your original value.