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

[Tim Daly]

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

issues.

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.

Tim

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