Re: Constructors motion
Alexandre Goldsztejn wrote:
If I remember well, Mr. Hayes has been arguing in last year
discussions that modal interval can be quicker on some very specific
problems. But I did not see anything concrete, although Arnold
Neumaier has spent lot of energy to try clarifying this question!
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.
In any case, regardles what P1788 decides to do (and for whatever reasons,
scientific or not), I'd like to put to rest a few things, ok?
Despite much of the accusations and misinformation that has been presented
by several individuals on this topic, there are a few facts that remain:
-- All of Sunfish Studio's patent applications are listed in the
references of the above-mentioned paper, and we have also complied with
IEEE's regulations and filed the appropriate paperwork with Baker on the
subject.
-- Without changing basis, there is no other mathematical alogirthm 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" (like Sederberg, etc.) require as much as 50X more operations. The
idea that such algorithms are just as fast is absurd, let alone the idea
such algorithms can be implemented efficiently in hardware.
-- Optimal bounds simply cannot be computed on the linear interpolation
operation using classic interval arithmetic without doubling the amount of
work. 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)
this certainly isn't any argument in favor of classical interval arithmetic
because a classical interval processor is still useless in this case.
-- 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.
Sincerely,
Nate