Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Descriptor system — where it appears

A linear system written as E x' = A x + B u with E possibly singular, so its transfer function is C(sE - A)^-1 B. The Loewner realisation returns one, and projecting it to the right order is what makes the pencil regular.

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 λ = 0λ = ∞λ = ∞-0.37690.5654.3836.4282 eigenvalues here, and it is one placecounted exactly, in rationalsfinite eigenvalues4at infinity2degree of det(A − λB)4worst residual, either kind5.5·10⁻¹⁴an eigenvalue is a ratioand a ratio has a direction, not a size

An eigenvalue with no value

If the second matrix of a pencil is singular then some of the eigenvalues are infinite, and that is not a degeneracy — it is the algebraic constraints of the model, one per constraint. What survives is a pair of numbers rather than one, and on the line those pairs live on, infinity is an ordinary point with an ordinary residual.

spectra · pencil

Named alongside it

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

Adaptive interpolationData driven realisationDeterminantExact arithmeticGeneralised eigenvalue problemInfinite eigenvalueInterpolationLoewner matrixMatrix pencilMcMillan degreeNoise floorNumerical rank

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