Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Interpolation — where it appears

Requiring an approximation to agree with what it approximates exactly at stated points. It is a constraint rather than an accuracy, so an interpolant of corrupted data reproduces the corruption and reports no error at all.

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

12345678910111210⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10¹order of the reduced modelworst relative errora model made of measurementssamples used24degree read6gap at the cut2·10⁸best model, order12its error2.7·10⁻¹¹states in the original40filled: at its own samplesopen: everywhere else

A model with no matrices behind it

Twenty-four numbers — values of a transfer function at twenty-four points — produce a sixth-order model of a forty-state system that passes through every sample to 10⁻¹² and matches the function it never saw to 10⁻⁹. Noise of 10⁻¹⁰ on those numbers takes the rank decision's gap from 10⁸ to 4.

reduction · data driven realisation 081624324000.250.50.7511.25angle of the strong direction (degrees)ratio, and convergence factorenergy ratio of oneenergy ratioclassicalminimiserwith the constraintwhere the formula is optimalenergy ratio at 0°1energy ratio at 45°1.2classical rate at 45°0.35the minimiser's rate there0.28a ratio of exactly one while the assumption holdsand a diagnostic when it stops

The formula that was already optimal

Ask for the interpolation that minimises the energy of its own columns and the answer is the classical AMG formula — to zero at every row of the one-dimensional Laplacian, and to four digits in two dimensions. On the operator rotated to 45° the two part company, and the gap between them is a diagnostic that needs no reference solution.

iterative · algebraic multigrid

Named alongside it

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

A normAdaptive interpolationAlgebraic multigridAnisotropyCoarse grid correctionConstrained minimisationData driven realisationDescriptor systemEnergy minimisationGalerkin operatorLoewner matrixMcMillan degree

All concepts