Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Stagnation — where it appears

The failure of a running sum to grow once each new term is below half the spacing of the numbers near its total. Every later term then rounds away, so the sum settles at a value set by the format's precision rather than by the data, however well conditioned the sum is.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

02505007501000250275300325350375additionsrunning totalround to nearest: nothing arrivesexactstochasticnearesta thousand additionshalf an ulp at 2561moves, round to nearest0moves, stochastic46relative error, nearest0.28relative error, stochastic0.0228 significand bits, unbounded exponenta flat line is not a small error

A coin flip that fixes the average

Add 0.1 to 256 a thousand times at eight significand bits and the answer is 256. Not approximately — the total never moves, not once, and no error bound says so. Round up one time in twenty instead of never, and it arrives at 348 against a true 356.

arithmetic · Rounding 0510152025303540first barrier depth at which the far well is wrong by half, octavesfp16the band adds 11 octaves · starting at the top adds 15E4M3the band adds 2 octaves · starting at the top adds 10E5M2the band adds 3 octaves · starting at the top adds 14flush to zero13gradual underflow24flush, started at the top29gradual, started at the top39flush to zero5gradual underflow7flush, started at the top13gradual, started at the top17flush to zero13gradual underflow16flush, started at the top27gradual, started at the top30dot: median of nine chains · bar: their rangethe headroom is worth more than the band

The far well in a byte

Half precision's subnormals carried a two-well chain's far well ten octaves deeper than flush to zero, and the essay that measured it predicted that on the eight-bit formats the band would be worth almost nothing and starting high almost everything. Measured on nine chains, the band is worth what it always was, two or three octaves; starting at the top of the range is worth ten and fourteen; and at three significant bits the far well is lost on eight chains of nine with no exponent limit at all, because the precision fails before the range does.

arithmetic · Subnormals

Named alongside it

The objects these essays reach for when they reach for this one.

SummationUnit roundoffError accumulationExponent rangeFlush-to-zeroFp8Gradual underflowHalf precisionRounding modesSignificandStationary distributionStochastic rounding

All concepts