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[Axiom-developer] Future research directions (Universal Numbers)

From: Tim Daly
Subject: [Axiom-developer] Future research directions (Universal Numbers)
Date: Mon, 7 Nov 2016 04:32:31 -0500

Numeric computations have been locked into IEEE arithmetic
for a long time. The problems are well known. NAG's numeric
routines have a huge number of parameters to deal with these

Axiom now has (growing) support for BLAS/LAPACK in native
form. This gives us the opportunity to explore alternative
number formats. BLAS routines could be adapted to use
interval arithmetic but initial experiments show that this tends
to generate wide, uninformative intervals.

Gustafson[0] has proposed a universal number format (UNUMs)
which essentially allows two things. First, you can re-define the
width of the exponent and mantissa fields to match the problem.
Second, you can define exact intervals to generate exact results.

Efforts are underway to create a new UNUM number format which,
hopefully, can be used in any context that DFLOAT can be used
(e.g. in polynomials).

In the longer but still near term Intel has bought Altera so it now
owns about 45% of the Field Programmable Gate Array (FPGA)
market. They have used their manufacturing to shrink the die size
by many generations and incorporated the FPGA onto the latest
Intel processor chip. What this likely means is the ability to define
new, novel instructions "on the fly" from your program. That would
allow "hardware UNUM support".

Ultimately this has the potential to allow strong symbolic/numeric
computation that does not suffer from overflow/underflow problems
and allows "symbolic epsilons". These symbolic epsilons can
eliminate the rounding-error / cancellation problems, leading to
more exact arithmetic and dynamic error analysis.


[0] Gustafson, John "The End of Error" 13-978-1-4822-3986-7 (2015)

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