Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Data-driven realisation — where it appears

Building a state-space model from samples of a transfer function alone, with no state matrix as input. The model interpolates every sample exactly by construction, so its interpolation error reports success whatever the data's own error is.

Named by 3 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 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 eight polesorder 5: true ÷ σ₆/σ₁457order 5: true ÷ midpoints1.712345610⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹order of the modelerror, over the largest |H|true errorerror at the midpointsmisfit at the samplesσₖ₊₁ ÷ σ₁ of the pencildotted: where the arithmetic stopsthe singular value is optimistic

An error estimate made of samples

A Loewner model built from samples of a transfer function comes with no error bound, and the number every code reads to choose its order — the pencil's next singular value — is the obvious stand-in. On four systems sampled at 48 points it is the error of the order-k model for the first two orders, within a factor of 2.5, and then falls away from it by about a decade an order: by the fifth it is optimistic by 109, 181 and 457, and on one system by 6,700 at the sixth. An AAA fit of the same degree to the same samples has an error within a factor of six of the Loewner model's at every order, so the shortfall is not the projection's. The model's misfit at its own samples is optimistic the same way. What works is more samples: the error at the 47 midpoints between the samples is within a factor of 2.2 of the true error at every order on every system, and the four midpoints nearest the poles alone within a factor of three.

reduction · Data-driven realisation

Named alongside it

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

Loewner matrixAdaptive interpolationInterpolationMcMillan degreeRational approximationTransfer functionApproximation before linearisationBarycentric formBranch pointDescriptor systemError estimateGreedy algorithm

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