Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Adaptive interpolation — where it appears

Choosing an approximant's interpolation points by its own residual rather than by a rule, one at a time. It locates a singularity nobody named and clusters at it geometrically, and past the noise level it chases the noise instead.

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

Also named here as data driven realisation — the same set of essays touches all of them, so they are one junction rather than several.

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 12345678910111210⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹degree of the approximantworst relative error on the target setchosen by the residualdegree reached12error there10⁻¹²even support, same degree1.1·10⁻⁷advantage702nearest support point0.096clustering ratio1.5filled: points chosen by the erroropen: points spread evenly

The points the algorithm chose

A rational approximant whose support points are picked by its own residual clusters geometrically at a branch point nobody named — recovering by measurement the rule a hand-built approximant is given. At degree ten it is seven hundred and fifty times more accurate than the same form with its points spread evenly.

polynomial · adaptive interpolation

Named alongside it

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

Data driven realisationLoewner matrixApproximation before linearisationBarycentric formBranch pointDescriptor systemGreedy algorithmInterpolationMcMillan degreeNoise floorRank is a decisionRational approximation

All concepts