Hankel singular values — where it appears
Named by 8 essays across one field — each of them below, with the objects they name alongside it.
The bound that is known in advance
Almost every error in numerical linear algebra 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.
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 one number predicts — 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.
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.
A condition number that is not the model's
κ(P)κ(Q) is quoted as the reason one route to the Hankel singular values fails, and it is. It is also read as a measure of how reducible a model is, and it is not — the norm of the Gramian does not move at all as the McMillan degree runs from four to fourteen, and the condition number wanders over a factor of thirty-five with no trend.
The state that is removed is not a mode
Balanced truncation removes one state and pays exactly twice one Hankel singular value. The natural reading is that the state removed is the model's least important mode and that the σ is that mode's own size — and on five systems that reading over-estimates by between 1.21 and 7.46, never once under.
A mode that rings is counted twice
Balanced truncation's bound, twice the sum of the discarded Hankel singular values, was attained at every order on the heat model — a formula for the error rather than a bound on it. On six lightly damped oscillators it is loose at every order but the last, by factors between 1.3 and 5, at every damping from 0.7 to 0.01. The reason is visible in the singular values: a mode that rings contributes two of them, nearly equal, and removing the mode costs about what removing one would. The last mode's bound over its error is 2.02, 2.00 as the damping falls to 0.03 and 0.01. The equality was a property of modes with one singular value each.
Where a ringing mode stops being a pair
A lightly damped mode puts two nearly equal Hankel singular values into balanced truncation, and the reduction error was read as twice the larger of the first pair cut — right to two per cent at damping 0.01, wrong by half at 0.1. Read instead as the pair's two values added, the error is right to within six per cent at every order up to damping 0.05, and the pair's split turns out to be twice the damping. What it cannot do is say when the reading fails. The split predicts nothing: the widest split at damping 0.1 has the best reading. The failures come from somewhere the pair does not look — the other dropped modes' response under the peak, and a cut between two singular values that agree to under one per cent, which left a reduced model with a pole a thousand times closer to the axis than any pole of the system it came from.
A tie is a gap smaller than its neighbours
A balanced truncation that cut between two Hankel singular values agreeing to under one per cent left a reduced model with a pole near the axis and twice the error the σ promised, and the warning drawn from it was a rule: never cut inside a tie. Over 1,452 reductions of twelve oscillators, all ten doubled errors are at ties — but three of them have gaps of about two per cent, which a one-per-cent rule lets through, and what they share is a gap under a fifth of the gaps beside it. Moving every such cut one order down never doubles and halves each doubled error, for a median cost of one per cent. It is not free everywhere: at damping 0.5 it costs up to two and a half times. And the near-integrator was never the harm: 32 reductions have one and 22 of them read their σ correctly. The cheapest test of all is one evaluation of the error at zero frequency, which picks out the ten doubled cuts and nothing else.
Named alongside it
The objects these essays reach for when they reach for this one.
Lyapunov equationBalanced truncationGramianA-priori boundSingular valuesTransfer functionCondition numberHinf normMcMillan degreeReduced stabilityUnit roundoffADI iteration