Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Exact ground truth — where it appears

A problem whose answer is known before anything is computed, because it was constructed from it, so an error is measured rather than estimated.

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

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.

error · scaling grey: proved empty · filled: proved to contain exactly one roota covered squareboxes proved empty42boxes proved unique2undecided0operator evaluations87every rectangle carries a proofand the two crosshairs are where the roots are

Where the box is cut

A branch-and-bound with an interval operator settles a whole square — two roots proved unique, forty-two regions proved empty, nothing left undecided, in 87 evaluations. Move the roots so one lands on the first bisection and it proves nothing at all, at any depth. Cutting at 0.485 instead of 0.5 finds both, in a quarter of the work.

arithmetic · interval

Named alongside it

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

BisectionBranch and boundComponentwise condition numberCondition numberEquilibrationExistence and uniquenessForward errorHilbert matrixInterval arithmeticKrawczyk operatorPerturbationPruning

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