Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Scaling — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 0246810110²10⁴10⁶10⁸10¹⁰10¹²spread of the row units (decades)condition numberκ_∞(DA)cond(DA)Hilbert κ_∞Hilbert condone system, two numbersκ_∞ at no spread9.8κ_∞ at 10 decades1.9·10¹⁰cond, either end7Hilbert, equilibrated1.3·10¹⁰the solution is the same at every spreadand one of these curves knows it

    The units the matrix is measured in

    One linear system, written twice. The rows of the second are the rows of the first in different units, the solution is identical to the last bit, and the condition number has moved by eight orders of magnitude. One of those two numbers is a fact about the problem and the other is a fact about the notation.

    rung 1 · error
  2. 10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1relative size of the entrywise perturbationrelative forward errorκ_∞ · εcond(A,x) · εmeasuredboth bounds holdκ_∞(A)1.9·10⁸cond(A, x)4.8ratio of the bounds4·10⁷both curves above the data are boundsand only one of them is a measurement

    A condition number scaling cannot move

    Skeel's componentwise condition number is invariant under any row scaling — exactly, before any norm is taken, because two diagonal factors cancel entry by entry. It is never larger than the normwise one and can be arbitrarily smaller, and the ratio between them is a diagnostic for which kind of ill-conditioning a matrix has.

    rung 2 · error

All ladders