Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Markov parameter — where it appears

The coefficients CAᵏB of a transfer function's expansion at infinity, one for each power of the state matrix. They are what a linear model returns to an impulse, read off one step at a time, and the rank of the Hankel matrix built from them is the McMillan degree. For a discretised operator they grow geometrically, at a rate set by the norm of the state matrix.

Named by 2 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 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

Named alongside it

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

Exact ground truthMcMillan degreeNumerical rankTransfer functionCondition numberDynamic rangeEquilibrationLoewner matrixOverflowRankSingular values

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