Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Balanced truncation — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    rung 1 · reduction
  2. 1357911131510⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1index kσₖ ÷ σ₁σ₁√utwo factorstheir productthe normal equations, againroutes agree to k =6product floors at9.7·10⁻¹⁰σ₁√u2.3·10⁻⁹κ(P)κ(Q)1.8·10³⁶√ of it1.3·10¹⁸do not form the productthe σ below the line are the bound

    The product nobody had to form

    The Hankel singular values are the square roots of the eigenvalues of PQ. Form that product and half of them stop existing, at a floor this site can predict from one number — and the fix is the one the least-squares field has had since its first essay, arriving in a place with no least-squares problem in it.

    rung 2 · reduction

All ladders