Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

McMillan degree — where it appears

The number of poles a transfer function actually has, which is the size a reduced model must have to reproduce it exactly. It is a property of the input–output map rather than of the coordinates, and it can be arbitrarily far below the state dimension.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

10⁻¹110¹10²10³10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹frequency ω|H(iω)|one function, two routesstates24relative, ω ≤ 10010⁻¹⁴against the terms3.9·10⁻¹⁵‖H‖∞0.0924 statesone function of one variable

A model that is a rational function

A state matrix has a hundred thousand rows and the thing anyone wants from it is a function of one complex variable. The number that says how much of that size was ever the complexity is a rank — and the rank a derivation writes down cannot be computed, while one built from samples alone can.

reduction · Transfer function 123456710⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹order r kepterror and bounds2Σσ, and the errorσᵣ₊₁a bound that is an equalityorders7bound ÷ error, worst1spread over the sweep1error ÷ σᵣ₊₁, worst2.1computed before the modeland attained by it

The bound that is known in advance

Almost every error in numerical linear algebra is measured after the fact. Balanced truncation has one that is computable before the reduced model exists, in a norm of a function rather than of a residual — and on ordinary problems it is not an upper bound that is loose. It is attained.

reduction · Balanced truncation 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 condition numbers, on a logarithmic scaleκ(P)5.66·10¹⁸κ(Q)5.71·10¹⁸√(κ(P)κ(Q)) — the SVD route5.68·10¹⁸κ(P)κ(Q) — the product route3.23·10³⁷1/u4.5·10¹⁵the mean, or the productand only one of them fits

A condition number that is not the model's

κ(P)κ(Q) is quoted as the reason one route to the Hankel singular values fails, and it is. It is also read as a measure of how reducible a model is, and it is not — the norm of the Gramian does not move at all as the McMillan degree runs from four to fourteen, and the condition number wanders over a factor of thirty-five with no trend.

reduction · Gramian conditioning 0246810121410¹⁵10¹⁹10²³10²⁷10³¹10³⁵10³⁹10⁴³k|CAᵏB|6 exact zerosa definition that will not computestates16exact zeros6‖A‖₂, measured1146last step-to-step ratio1330range across the run2·10²⁹zero for the travel timethen ‖A‖₂ᵏ

The definition asks for more of what defeats it

The rank of a p × p Hankel matrix of Markov parameters resolves a degree of p, and it needs 2p parameters to do it. Those parameters grow like the norm of the state matrix raised to their index, so the count that buys resolution is the same count that buys dynamic range. One model, four run lengths, and a spread that runs from 10¹¹ to 10¹⁰².

reduction · Transfer function ‖H − Hₙ₋₁‖∞ ÷ σₙpositive residues2.0000mixed residues2.0000clustered poles2.0000four poles2.0000wide spectrum2.0000‖ΔH‖ 3.89·10⁻⁴ σ 1.94·10⁻⁴‖ΔH‖ 3.92·10⁻⁴ σ 1.96·10⁻⁴‖ΔH‖ 0.00502 σ 0.00251‖ΔH‖ 8.41·10⁻⁴ σ 4.21·10⁻⁴‖ΔH‖ 4.6·10⁻⁴ σ 2.3·10⁻⁴2four systems, four ratios, one number: 2.0000one state, one equalityand the sum is that, repeated

The state that is removed is not a mode

Balanced truncation removes one state and pays exactly twice one Hankel singular value. The natural reading is that the state removed is the model's least important mode and that the σ is that mode's own size — and on five systems that reading over-estimates by between 1.21 and 7.46, never once under.

reduction · Balanced truncation 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.

Transfer functionSingular valuesBalanced truncationExact ground truthGramianHankel singular valuesLoewner matrixLyapunov equationNumerical rankA-priori boundCondition numberData-driven realisation

All concepts