Re: Suggestions for hardware implementation

[Ulrich Kulisch <Ulrich.Kulisch@xxxxxxxxxxx>]

Please have a look at the Kirchner/Kulisch paper. I attach a copy

Best wishes
Ulrich


Arnold Neumaier schrieb:
> Section 1.8 of ViennaProposal-v3.01 reads:
>
>
> 1.8. Suggestions for hardware implementation
>
> A proposed minimal useful set of hardware operations which would
> significantly speed up existing applications in constraint programming,
> global optimization, and computational geometry are the following
> operations defining core interval arithmetic.
> - forward and reverse interval operations for
>   plus, minus, times, divide, sqr (Sections 3.8-3.11 and 7.9)
> - mixed operations with one interval argument for
>   plus, minus, times, divide (Section 4.2)
> - forward sqrt, exp, log (Sections 3.8-3.9)
> - linearInt, linearExt, linearInv (Sections 5.8 and 7.11)
> - mid, rad (Sections 5.2 and 7.11)
> - division with gaps (Section 5.7)
> - optimal enclosure of sums and inner products (Section 5.1)
> These operations might be singled out to represent (together with the
> operations from Section 2.5) level 1 of the standard; the remaining
> operations would represent level 2.
> An appendix may suggest explicit model algorithms for all level 1
> operations.
>
> [end of Section 1.8]
>
>
>
> I'd like to discuss the following additional suggestion for hardware 
> inplementation:
>
>
> What if we propose implementing three different pipelines for
> floating-point operations in fixed, selectable rounding modes?
>
> This can be used in ordinary floating-point calculations to
> parallelize computations, and in interval calculations to do operations
> in different rounding modes in different pipelines.
>
> This would completely remove the penalty for rounding mode switches;
> an optimizing compiler can choose the rounding modes of each pipeline
> to maximize total throughput with a minimal number of switches.
> For example, during interval computations interspersed by
> approximate floating-point computations (frequent in global 
> optimization), one pipeline might be set to round up, one to
> round down, and one to round nearest.
>
> Note that there are many situations (for example, the Hansen-Bliek
> method for solving linear systems with interval coefficients) where
> one wants to do some directed rounding calculations within some routine
> using interval arithmetic; so the directed pipelines should be
> available for both interval and other directed calculations.
>
> Other applications of directed rounding arise when explicit use is
> made within interval computations of monotonicity behavior of certain
> formulas, which allows to compute lower and upper bounds separately
> using directed rounding. A well-known example is Barth and Nuding's
> optimal solution of linear systems with an interval M-matrix, but there
> are many other cases of interest; e.g. the algorithms in Section 5.8 of
> ViennaProposal-v3.0.
>
>
> Arnold Neumaier

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