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Is it possible to get a response to
Baker's questions? In order to discourage further discursive
description in which the answer may or may not be present, Let me
try to rephrase as
yes/no questions:
Given a two vectors A,B of length n , will there be a difference
in any bit in the answer
between double_correctly_rounded_dot_product(A,B,n)
and reduce_to_double(exact_dot_product(A,B,n))?
yes or no?
Does the requirement of the exact dot product require a new
representation consisting of an
array of a certain length d of base b numbers?
yes or no?
I add the question: if the previous answer is yes, are we now
involved with proposing
the definition , standardization, and a complete repertoire of
heretofore new operations
in the interval standard, operations which in general do not have
interval arguments or interval results?
yes or no?
I would be happy to share programs to compute exact as well as
correctly rounded
dot products which do not use complete arithmetic (CA).
Therefore
it is certainly not the case that CA is required for EDP. Sparse
representations
have advantages, and the whole idea is somewhat redundant in a
language that
has as a base type exact rational and integer arithmetic.
In the interest of producing a standard document and at least one
reference implementation
before the IEEE deadline, I think other issues which might need
discussion should be raised.
Or is everything set and this is the last area that needs
discussion?
um, yes or no?
Thanks
RJF
On 9/5/2013 12:27 PM, Ulrich Kulisch wrote:
Am 05.09.2013 13:59, schrieb Ralph
Baker Kearfott:
Can someone please clarify or confirm to me how we view
the EDP? In particular, the only practical difference I see
between the EDP and a correctly rounded dot product
(the present state of P-1788's plan) is that a possible
EDP operation must return some representation of
the exact result (not a standard floating point number),
whereas the correctly rounded dot product must return
a floating point number the same AS IF the dot product
were first computed exactly, then rounded according to
the rounding mode in effect.
Thus, if my interpretation is right, the only additional requirement
of the EDP over the correctly rounded DP is that a representation
from which the exact result can be recovered be available. Wouldn't
whether or not that result is subsequently rounded into a floating point
number then be up to the programmer or user?
Baker
Baker,
Let R = R(b, l, e1, e2) be the
floating-point system, b the base, l the number of matissa
digits.
A variable of type "complete" would be a fixed-point
variable with d = t + 2•e2+2l+|2•e1|
digits of base b. For n<= bt , every sum of n
floating-point products can be represented as a variable of
type complete. Moreover, every such sum can be computed in a
(local) store (or register) of length d without loss of
information.
If A is a variable of type complete and x and y are
floating-point numbers, the function call
dotadd(A,
x, y) or the assignment A := A + x•y adds the double length
product x•y to the variable A exactly. The EDP of two
vectors x = x[i] and y = y[i] is now easily implemented with
a variable A of type complete as follows:
A := 0;
for i:= 1
to n do dotadd( A, x[i], y[i]);
z := ○(A);
The last statement then rounds the value of
type complete (to nearest, upward, downward, or to the least
including interval depending on the rounding symbol o) to
the type of z. Here z could be of any higher precision.
For example the method of defect correction
requires highly accurate computation of expressions of the
form
a•b
+ c•d + e•f with
vectors a, b, c, d,
e, f. The result
involving 3n multiplications and 3n -1 additions is produced
with no roundings after the additions and multiplications.
Digit cancellations as in a conventional floating-point
computation do no occur here. Then the exact result can be
rounded to a desired precision.
Best regards
Ulrich
P.S.: I shall be out of town
for a couple of days.
--
Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik
D-76128 Karlsruhe, Germany
Prof. Ulrich Kulisch
Telefon: +49 721 608-42680
Fax: +49 721 608-46679
E-Mail: ulrich.kulisch@xxxxxxx
www.kit.edu
www.math.kit.edu/ianm2/~kulisch/
KIT - Universität des Landes Baden-Württemberg
und nationales Großforschungszentrum in der
Helmholtz-Gesellschaft
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