Re: re motion 50 (Just to you...)

[Ulrich Kulisch <ulrich.kulisch@xxxxxxx>]

Let's assume that we are using decimal arithmetic. Consider the two vectors
a = (10²ºº, 23456, -10²ºº)   and b = (1, 1, 1).
A conventional for loop for the dot product of a and b in any available 
fixed precision gives you 0, while the EDP would give you 23456.
A conventional for loop for the dot product of a by itself (norm) would 
give you an overflow, while the EDP would give you the value of the 
product.23456×23456 with no overflow.

For more details see the poster.

Let me mention that I shall be out of town for a conference for one week.

Ulrich Kulisch


Am 17.09.2013 23:09, schrieb Richard Fateman:
> On 9/17/2013 6:24 AM, Ulrich Kulisch wrote:
>
> > Let me just discuss an explicit example more closely, computing the
> > dot product of two vectors with interval components. What you would
> > like to have is the least enclosure of the set of all dot products of
> > real vectors out of the two interval vectors. Computing the interval
> > dot product in conventional interval arithmetic (what we are going to
> > standardize in P1788) for each interval product of two vector
> > components you round the minimum of all products of the interval
> > bounds downwards and the maximum upwards.
> There is no requirement that an interval dot product be computed by a
> simple loop
> for i:=1 to n sum a[i]*b[i]      {where sum and * are interval operations}
> just as there is no requirement that a dot product of floats be computed
> by that same loop.
>
> If I were computing a dot product of vectors of ordinary floats I might
> consider
> extra-precise multiplication (via Split/TwoSum/TwoProd   etc.)
> and compensated summation.
>
> For the analogous interval operation, perhaps the  convenient operations
> I would need
> are already implicit in the standard, which permits
> multiple-precision.... For example
> multiplication of 2 double-float intervals [a1,a2] * [b1,b2]  to produce
> [C,D]  where C and D were
> quad-float numbers. e.g. C = <e,f>  where e + f ,each a double-float, is
> a representation of exactly the product.
>
> This would be available as an appropriately overloaded interval mul(),
> with a quad target precision, e.g.
> quad_mul(a,b).
>
> I think that quad_add() would be effective in adding the minima and the
> maxima, vastly decreasing the
> possibility of a significant rounding error affecting the final outcome.
> Or perhaps a compensated summation of the collection of (scalar) values
> separately.
>
> While it is possible to add 3 numbers a,b,c  via  EDP(<a,b,c>, <1,1,1>)
> and multiply two numbers by EDP(<a>,<b>), it does not seem economical.
>
> RJF


--
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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