Re: CA and EDP
Playing with Maxima (Macsyma?), Richard Fateman found
> ... Using complete arithmetic with 4288 bit fractions (about 1290 decimal
> digits) seems to be overkill for this iteration at x[2790] for a scalar
> result. Using 1290 decimal digits I can get reasonable answers much
> further out than 2790.
Funny; I did similar experiments using EXEC-3 (a modern version of EXEC-2).
Using 644 digits and comparing to a 1000-digit run until the two started
to differ in the two high-order digits, I ran through 4100 iterations.
I picked 644 because the square of the smallest binary64 is about 1e-644.
The first 17 digits (nominal precision of binary64) matched until 3980.
But the point remains: results of the logistic iteration simply match
the available precision; there's nothing magical about CA here; on the
contrary: CA is fixed-precision machinery. Up to its capacity it can
have a speed advantage however -- IF it is available in hardware.
I have a more serious question however. A correctly-rounded dot product
can be used to distinguish a singular matrix from a badly ill-conditioned
one (exact zero determinant vs inexact zero due to final underflow). Does
an exact dot product provide more information to help deal with this? It
seems to me that a correctly-rounded dot product with an extended-precision
reslt, or one with underflow exponent wrap-around, or with a scaled result,
would provide the same information. (A scaled result would overcome the
limitations of a bounded exponent range.)
Michel.
---Sent: 2013-08-08 03:36:58 UTC