Re: Information

[Ian McIntosh <ianm@xxxxxxxxxx>]

For intervals, shouldn't EDP be implemented as returning an interval - the smallest interval enclosing the exact result?  For that, you don't really want to produce a single correctly rounded DP then use that to construct the interval.  You want to round down and up directly into an interval.

- Ian McIntosh          IBM Canada Lab         Compiler Back End Support and Development


David Lester ---09/06/2013 07:20:39 AM---On 6 Sep 2013, at 10:56, J. Wolff von Gudenberg wrote: > No Dave,

From:	David Lester <dlester@xxxxxxxxxxxx>
To:	Ian McIntosh/Toronto/IBM@IBMCA
Date:	09/06/2013 07:20 AM
Subject:	Re: Information

---

On 6 Sep 2013, at 10:56, J. Wolff von Gudenberg wrote:

> No Dave,
> your answer is wrong


Wolff,

Granted, but you have to admit that it ensures we have something concrete to focus on.

Dave.

ps Thanks for the correction.

> Am 06.09.2013 08:30, schrieb David Lester:
>> Richard,
>> 
>> Let us see if I can answer your questions:
>> 
>> On 6 Sep 2013, at 04:24, Richard Fateman wrote:
>> 
>>> 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?
> NO !   the Definition of correcty rounded  is : complete exact result (of dotproduct) and then round
>> Yes.
>> 
>> Suppose A, and B are of dimension 3, with a_1*b_1 = 10e350, a_2*b_2 = -10e350,
>> and a_3*b_3 = 1e-350.
>> 
>> (Substitute an alternative to 350 in the above if I've got the wrong exponent.)
>> 
>> The exact dot product will get 1e-350.
>> 
>> Depending on how it's implemented double_correctly_rounded_dot_product might get any number of
>> answers, but 0 is quite likely --- even if quad precision is used internally.
>> 
>> 
>> 
>>> 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?
> YES but not necessarily an array or fixed point number
>> Internally yes. Externally no.
>> 
>> Internally we will need to hold an integer base b of length 2 * 2^max_exponent. Whatever that is in quad precision.
>> 
>>> 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?
> in a certain sense YES
>> Yes. Indeed I'd go further and say that by itself EDP is merely one part of an interval DP, since we will of necessity have to represent the two end points of the interval enclosing the dot product.
>> 
>> cut
>> 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?
>> I fear the answer is a resounding "no"!
>> 
>> Dave.
> NO ! Jürgen
>>> 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
>>>> 
> 
> -- 
> -                Prof. Dr. Juergen Wolff von Gudenberg
>     o           Lehrstuhl fuer Informatik II
>    / \          Universitaet Wuerzburg, Am Hubland, D-97074 Wuerzburg
> InfoII o         Tel.: +49 931 / 31 86602 Fax ../31 86603
>  / \  Uni       E-Mail:wolff@xxxxxxxxxxxxxxxxxxxxxxxxxxx
> o   o Wuerzburg
>