Re: Continuations

On Mon, 22 Jul 2024 11:01:15 +0000, Michael S wrote:

On Fri, 19 Jul 2024 20:55:47 +0200
Terje Mathisen <terje.mathisen@tmsw.no> wrote:
>

MitchAlsup1 wrote:

On Fri, 19 Jul 2024 14:16:01 +0000, Terje Mathisen wrote:

Back when I first looked at Invsqrt(), I did so because an
Computation Fluid Chemistry researcher from Sweden asked for help
speeding up his reciprocal calculations
(sqrt(1/(dx^2+dy^2+dz^2))), I found that by combining the 1/x and
the sqrt and doing three of them pipelind together (all the water
molecules having three atoms), his weeklong simulation runs ran in
half the time, on both PentiumPro and Alpha hardware.

>
I, personally, have found many Newton-Raphson iterators that
converge faster using 1/SQRT(x) than using the SQRT(x) equivalent.

>
Yeah, that was eye-opening to me as well, to the level where I
consider the invsqrt() NR iteration as a mainstay, it can be useful
for both sqrt and 1/x as well. :-)
>
Terje
>

>
What is this "SQRT(x) equivalent" all of you are talking about?
I am not aware of any "direct" (i.e. not via RSQRT) NR-like method for
SQRT that consists only of multiplicationa and additions.
If it exists, I will be very interested to know.

There are certain N-R iterations that can be expressed with both::
     NR+1 = F( NR,  SQRT() )
and
     NR+1 = F'(NR, RSQRT() )
Typically the one with RSQRT() converges slightly faster than the
one using SQRT(). How much is slightly::maybe ½-1 more bit per
iteration.