Numerical Linear Algebra
where the algebra stops being the arithmetic
Series

Data-driven realisation — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · reduction
  2. 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.

    part 2 · reduction
  3. 0.0010.010.1110100100010⁴next singular value0 samples4 midpoints near the poles4 samples47 midpoints47 samplesnext order, no sample0 samples4 points it chooses4 samplestrue error over the estimate: right of one, the estimate is optimisticone dot per order and systemfour chosen points are the error

    A pole between the samples

    A Loewner model needs a check on its error, and the one predicted to be cheap is four points chosen where the model disagrees most with the model one order higher. With exact samples it is better than predicted: on four systems the true error at those four points is the model's error within two per cent at every order, and the disagreement alone, with no sample at all, within three per cent at every order but the last. With noisy samples the checks keep working on every model that has no pole between the samples, and seventeen of thirty-two models do. Past the order the noise allows, the projection puts a real pole inside the sampled interval, so the error is infinite at one point and small a few per cent of a gap away, and every finite set of checks reports a number between 130 and 49,000 times too small. The test that catches every such model costs nothing: the model's own poles.

    part 3 · reduction

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