Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Regression tolerance — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. -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.

    rung 1 · machine
  2. 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.

    rung 2 · machine

All ladders