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 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 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 on this site 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

Named alongside it

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

Exact ground truthSingular valuesTransfer functionA-priori boundBalanced truncationCondition numberGramianHankel singular valuesLoewner matrixLyapunov equationMarkov parameterMatrix equation

All concepts