Numerical Linear Algebra
where the algebra stops being the arithmetic
Field

Reduction, and what a model is for

A state matrix is n × n and the thing anybody wants from it — the map from an input to an output — is a function of one complex variable. The gap between those two numbers is this field. A reduced model is an approximation of the function and not of the state, so its error is measurable as a norm of a function rather than as a residual, and one of the two ways of building it comes with a bound computable before the model exists. That bound turns out not to be an inequality: on ordinary problems it is attained. What makes any of it possible is that a Gramian's eigenvalues fall off a cliff, and the reason they do is a rational approximation problem with a closed-form rate.
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.

 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.

 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.

 10⁻¹110¹10²10³10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵s, on the real axis|H − Hᵣ| ÷ |H|exact where askedpoints4conditions bought8worst at a point5.3·10⁻¹⁶worst away from one8.4·10⁻⁴4 points, 8 conditionsand no bound in between

Exact at the points that were named

Balanced truncation asks for nothing and bounds everything, at a cost no large model can pay. The other kind of reduction asks for r numbers, costs r solves, is exact at every one of them — and bounds nothing anywhere else. That trade is the whole of large-scale model reduction.

 2345678110²10⁴10⁶10⁸10¹⁰vectors in the basisκ₂ of the basisas derivedsolves spread outone subspace, two spanning setseight moments at one point7.7·10⁹eight points, spread1growth per vector661/u4.5·10¹⁵the same subspaceand only one of them usable

A basis that is the same subspace and not the same thing

The interpolation conditions are conditions on a subspace, so any basis of it will do. The one a derivation writes down reaches a condition number of 7.7·10⁹ by its eighth vector, and the rate at which it gets there is set by a number the user chose with no information.

 01234567891011121310⁻¹110¹10²10³iterationinterpolation point σa fixed point that is a conditionorder4iterations13σ against −λ(Aᵣ)2.8·10⁻¹²H₂ error4.3·10⁻⁵it stopped movingand the condition holds there

Interpolating at the model’s own poles

One choice of interpolation points is not arbitrary — the mirrored poles of the model about to be built. It is a fixed point rather than a guess, and when it is reached it beats a method costing O(n³) — by 0.4 per cent, which is the honest size of the whole contest.

 1234567891010⁻¹⁹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹kλₖ₊₁ ÷ λ₁, and the boundZ_k²the Gramianpredicted from two numbersstates30κ of the spectrum389λ11 ÷ λ₁2.5·10⁻⁸the bound there5.2·10⁻⁴the cliff everything rests onand the reason for it

Why a Gramian can be truncated at all

Every method in this field rests on one fact nobody states the reason for — the eigenvalues of a Gramian fall off a cliff. The equation defining it has a rank-one right-hand side and no low-rank structure anywhere — and the answer's decay is a rational approximation problem with a closed-form rate.

 10¹10²10³10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹λ, over the spectrum of −A|r(λ)|geometric maxequally spacedgeometricone line of codeshifts6κ of the spectrum389geometric max0.12equally spaced max0.89the factor between7.5the same k solvesand one choice of where

Where to put the poles of a rational function

Three times in one field the same question arrives from different directions — ADI shifts, rational approximation of a square root, the decay of a Gramian — and it has one answer. Cluster them geometrically towards wherever the function is difficult, and the alternative that looks reasonable costs orders.

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