Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Rounding modes — where it appears

The rules a format offers for choosing between the two representable numbers straddling an exact result. To nearest is the default and makes errors cancel; always up or always down makes them add, which is the whole of interval arithmetic.

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

10²10³10⁴10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴terms summedrelative error+∞, −∞ 1.01zero 1.00stochastic 0.50nearest 0.47√n against n, fittednearest, fitted exponent0.47stochastic, fitted exponent0.5toward +∞, fitted exponent1twelve seeds averaged at each sizethe slope is the bias, not the precision

The direction the error leans

The size of one rounding error is set by the precision. How ten thousand of them combine is set by something else entirely — the rounding mode — and the fitted exponents are 0.47 for round-to-nearest and 1.01 for round-toward-infinity, on identical data at identical precision.

arithmetic · Rounding 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

Named alongside it

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

Error accumulationSummationUnit roundoffDirected roundingHalf-precisionIEEE 754Interval arithmeticStagnationStochastic roundingTruncationThe ulp

All concepts