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)
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