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