Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Regression tolerance — where it appears

The threshold a test applies when comparing a computed answer with a stored one. It has two sides: below the variation a correct build already produces it fails on correct work, and above the smallest defect worth catching it accepts the thing it exists to catch.

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

-15-13-11-910⁻¹³10⁻¹¹10⁻⁹10⁻⁷log₁₀ of the defect, relative to the entry it sits inhow far the answer movedthe machine, on its owna tolerance with two sidesmachine band3.2·10⁻¹²smallest caught10⁻¹²its nearest run4.6·10⁻¹²defects hidden3window, factor1.4below the band nothing is visibleand above it everything is

What a regression test can ask for

The machine's own variation on one solve is 3.2·10⁻¹², and the smallest defect whose answers clear it is one part in 10¹². The tolerance exists, it is bracketed on both sides by a factor of 1.42, and it is neither zero nor the 10⁻⁸ that usually gets typed.

machine · regression tolerance κ of the suma component of b − Ax8.09·10¹⁷ad − bc, near-degenerate3.6·10¹⁶qᵢᵀqⱼ, an orthogonality check3.26·10¹⁶zᵀAz, a trace probe95.7pᵀAp, a curvature73.7rᵀr, a residual norm1measured, not assumedhighest8.1·10¹⁷lowest1above 10¹⁰3terms128sums of squares are safeand nobody decides anything from one

Two machines, one certificate

Nothing a solver returns says which of its answers you got. Four things could be reported instead — the summation condition number, the partition count, an exactly accumulated residual and a directed-rounding interval — and each costs about one pass over data the routine already has in hand.

machine · regression tolerance

Named alongside it

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

Bitwise reproducibilityRun-to-run variationAlgorithm selectionBackward errorCertificateDirected roundingExact accumulationInterval arithmeticNoise floorReduction orderSilent failureSummation condition number

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