Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Reproducible summation — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. distinct answersone accumulator303eight pieces72compensated119pre-rounded1exact1worst error 1.17·10⁻⁸worst error 4.61·10⁻⁹worst error 4.96·10⁻¹⁰worst error 4.91·10⁻⁶worst error 0bitwise, or not at allpermutations400one accumulator303pre-rounded1its error4.9·10⁻⁶compensated error5·10⁻¹⁰accuracy and agreement are different propertiesand the accurate one is not the agreed one

    The sum that cannot be wrong

    Snap every addend to a common multiple before adding, and every partial sum is exact — so the order stops mattering, by construction rather than by luck. Four hundred permutations return one value where an ordinary reduction returns three hundred and three.

    rung 1 · machine
  2. 02468101210⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸binary64 operations per elementrelative errorruntime orderfixed partitionpre-rounded, one passpre-rounded, two passescompensated (Kahan)exact accumulationwhat determinism costsreproducible policies3cheapest, ops/element3its error2.3·10⁻⁷two passes10⁻¹⁷ordinary, 8 pieces2.7·10⁻¹¹nothing is cheap and exactthe middle of the figure is the answer

    What determinism costs

    Six ways to add up a vector, priced in operations per element and in accuracy. Nothing sits in the bottom left of the figure — an answer that is the same on every machine costs between three and twelve operations where an answer that is not costs one.

    rung 2 · machine
  3. 02468101210⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²binary64 operations per elementrelative errorruntime orderfixed partitionpre-rounded, one passpre-rounded, two passescompensated (Kahan)exact accumulationwhat determinism costsreproducible policies3cheapest, ops/element3its error0.0024two passes10⁻¹⁷ordinary, 8 pieces3.6·10⁻⁷nothing is cheap and exactthe middle of the figure is the answer

    Accuracy and agreement are different properties

    The most accurate policy on this site's summation figure returns 119 different answers, and the one that returns a single answer is four orders less accurate. Neither property implies the other, and the vocabulary has one word for both.

    rung 3 · machine

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