Re: Motion on interval flavors
Alexandre Goldsztejn wrote:
- Scientific point of view. Several experts, including me, have
claimed that every thing that can be done using Kaucher intervals can
also be done using standard intervals
Folks, these claims by Alexandre are simply false.
Chapter 6 of the paper:
http://grouper.ieee.org/groups/1788/Material/Hayes_Modal%20Intervals.pdf
explains a very important application of modal intervals (in hardware) to
calculating narrow bounds on Bezier curves, a mathematical problem that is
at the core of applications in the computer graphics and CAD industries. It
has been available publicly for several years.
Here are the facts:
-- Without changing basis, there is no other mathematical algorithm to
evaluate a Bezier curve in fewer arithmetic operations than de Casteljau.
Some classic algorithms that Arnold argues in one of his papers are "just as
good" require as much as 50X more operations. The idea that such algorithms
are just as fast, or that they can be implemented efficiently in hardware is
quite absurd.
-- Optimal bounds simply cannot be computed on the linear interpolation
operation using classic interval arithmetic without doubling the amount of
required operations. In this respect, a would-be IEEE 1788 interval
processor will always be slower than an equivalent modal interval processor
in these calculations.
-- Reverse-engineering the modal arithmetic into primitive
floating-point operations has several fatal (from a performance
persepective) flaws: (a) the required if-then-else branching means it can't
be computed on existing floating-point vector hardware, (b) the branching
can stall the processor to lead to excessive performance penalties, and (c)
the result is a bunch of floating-point operations, since a classical
interval processor is useless to compute the optimal bounds.
-- A modal interval processor is the simplest and most efficient
implementation in hardware to perform these calculations. It has the benefit
of also being a general-purpose computational tool that can be used for
other calculations, as well, and the recursive linear interpolation
operations in the de Casteljau algorithm can be deeply pipelined without any
penalty of branching to achieve the most optimal performance.
Nate
P.S. An FPGA prototype of a modal interval co-processor is already in
development here at Sunfish, and if things go well an ASIC version may be
introduced into the field sometime next year.