Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Directed rounding — where it appears

2 essays name this object, across one field. What follows is each of them, and the objects they name alongside it.

Also named here as interval arithmetic — the same set of essays touches all of them, so they are one junction rather than several.

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 10⁻⁹10⁻⁶10⁻³110³10⁶13212937κ · usignificand bitsκu = 1a bound was provedthe method refusedit never returns a wrong boundlargest κu with a proof0.45smallest κu without one0.89cases refused, of the grid13a refusal is not a wide bound — it is no bound at alland it is the only failure mode here

A bound that is proved

Every error statement on this site so far is a measurement of one run. Interval arithmetic makes a different kind of claim — the answer lies in this set, for this input, with no probability attached — and its failure mode is that it returns nothing at all. On a Hilbert system it proves a bound 23 times the error it bounds, and one size later it refuses.

arithmetic · interval

Named alongside it

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

Interval arithmeticUnit roundoffCancellationCondition numberError accumulationHilbert matrixIEEE 754Rounding modesSummationTruncationVerified computingWrapping effect

All concepts