Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Skeel condition number — where it appears

A condition number for a linear system measured componentwise: the norm of |A⁻¹||A|, optionally weighted by the solution. It is unchanged by row scaling, so it can be far smaller than the normwise κ(A) when the difficulty is only a matter of units.

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

-12-10-8-6-4-2010⁻¹⁷10⁻¹³10⁻⁹10⁻⁵10⁻¹10³10⁷10¹¹10¹⁵log₁₀ μcondition number, and relative errorκ₂, augmentedcomponentwise, condensedcomponentwise, augmentederror, augmentedone matrix, two numbersκ₂ at μ = 10⁻¹²3·10¹³componentwise, same matrix13their ratio2.3·10¹²measured error9.4·10⁻¹⁶every library prints the top lineand the error obeys the third

Two condition numbers of one matrix

κ₂ is a worst case over perturbations of a given norm, and a normwise perturbation may put its whole budget on the smallest entry. The componentwise number is a worst case over perturbations proportional to the entries, which is what a backward-stable factorisation actually makes. On one matrix they are 3·10¹³ and 13.3, and the error obeys the second.

error · Scaling deepest barrier, far well right to within halfno exponent limit40gradual, started at 102433fp16, gradual underflow23fp16, flush to zero13051015202530354010⁻⁵10⁻⁴10⁻³10⁻²10⁻¹110¹barrier depth, octaves below the near wellrelative error in the far well's masssmallest normalsmallest subnormalno exponent limitgradual, started at 1024fp16, gradual underflowfp16, flush to zeroan error of one: the far well is emptysubnormals bought nine octaves

The well on the far side of the band

Gradual underflow was said to buy a predicate and not an answer, because a quantity that has decayed into the subnormal range is already lost. A quantity that passes through the band on its way somewhere else is not. The stationary distribution of a two-well chain, computed in half precision across a barrier whose top is two to the minus sixteen of the near well, keeps its far well's probability of 0.2454 to three digits with subnormals and returns exactly zero without them. The normwise backward error calls both answers exact.

arithmetic · Subnormals

Named alongside it

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

Backward errorComponentwise condition numberBidiagonal matrixCondition numberExact ground truthExponent rangeFlush-to-zeroForward errorGradual underflowInterior-point methodSaddle-point systemsScaling

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