Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Linearisation backward error — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 0246810⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹log₁₀ γ, the change of unitsrelative errorforward errorη, the quadraticη, the linearisationagainst a closed formη(linearisation), worst7.6·10⁻¹³η(quadratic), worst1.2·10⁻⁴forward error, worst0.0013coefficient spread4.2·10¹⁵the solver is right at every stopabout a problem nobody asked

    A backward-stable answer to a problem nobody asked

    One quadratic eigenvalue problem, in nine systems of units, with a change of variable that is exact in both directions. The residual the solver prints stays at the rounding level at every stop. The answer loses eleven orders of magnitude, and the two facts are consistent.

    rung 1 · polynomial
  2. first · leading5.47·10⁻⁶first · trailing4.62·10⁻⁵second · leading4.27·10⁻⁵second · trailing2.23·10⁻⁴symmetric · leading5.47·10⁻⁶symmetric · trailing4.05·10⁻⁵all six are the same algebrabest route5.5·10⁻⁶worst route2.2·10⁻⁴spread across the six41condition of the linearisation8.3·10¹²the spectra agreeand the arithmetic does not

    Six routes to one spectrum

    Three linearisations of one quadratic, each reduced to a standard eigenvalue problem two ways. All six have exactly the same eigenvalues in exact arithmetic. On a well-scaled problem they differ by noise; on a badly scaled one by a factor of forty; and two of the six are the same matrix.

    rung 2 · polynomial

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