[
Date Prev][
Date Next][
Thread Prev][
Thread Next][
Date Index][
Thread Index]
IEEE754R13
Craig Nelson wrote on July 21:
In your 1st example, it strikes me that the input variables (b, A)
may (unlikely if they arise from a physical problem where data
are measured by instruments of finite precision) be precisely
known in decimal, but will be imprecise when converted to binary
floating point. To be correct, I think you would have to demand
that decimal arithmetic would be supported and utilized if the inputs
are decimal scientific notation..
YOUR EXAMPLE:
I give only two counterexamples for this assertion. I could easily give
more.
1. For a system of linear equations A x = b, many solvers repeatedly
compute the defect d = b A x' , where x' is an approximation of
the solution. In the expression for d all data, and x' in particular,
are exact. With an exact scalar product d can be computed exactly.
My COMMENT on this:
A computer can only solve problems for data that are in the
computer. These data define the problem. In this sense they are exact.
It is the user's responsibility to bring the data in the computer
with sufficient accuracy. For a particular problem you may well
think that the data do not describe your problem accurately enough
if they are read through standard input conversion or if they are
otherwise contaminated. Then it would be most natural to read the
data into the computer as intervals or to a higher precision and
solve the problem for these data with interval or higher precision
arithmetic.
Of course, nobody would do this. On existing processors it would
slow down the computation by orders of magnitude. The proposed IEEE
P754 arithmetic as it is presently specified will not change this
situation much.
The simple arithmetic suggested under CONCLUSION in my mail,
however, would change the situation dramatically. It provides interval
arithmetic at the speed of simple floating-point arithmetic and high
speed arithmetic for dynamic precision for real and interval data.
Best regards
Ulrich Kulisch
--
*******************************************************
Ulrich Kulisch
Institut für Angewandte Mathematik
Universität Karlsruhe
76128 Karlsruhe, Germany
Tel.: (+49) 721 608 2680 oder (+49) 721 608 4202
*******************************************************