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Toward accurate polynomial evaluation in rounded arithmetic

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If you have a question about this talk, please contact Grigorios Mingas.

This talk will present the work of Demmel et al, 2005, on the ‘accurate’ evaluation of polynomials. More precisely an algorithm is sought for a polynomial p and open real domain D such that for any desired relative accuracy < 1, there exists a machine precision such that executing the algorithm on hardware with that machine precision will return answers for all inputs in D which satisfy the accuracy requirement. Crucially, the existence of such algorithms depends on arithmetic components available. If only floating point addition and multiplication are available then no algorithm exists for accurate evaluation of:

z6 + x2 y2 (x2 + y2 – 4z2)

but there is for:

z6 + x2 y2 (x2 + y2 – 3z2)

The reason rests on considering the ‘algebraic variety’. Steps to making a ‘compiler’ which produces the algorithm or shows that none exists will be presented.

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