Each method wins the norm it was designed for, and by less than anyone guesses
At its defaults it draws each method wins the norm it was designed for, and by less than anyone guesses. Ratios of the loser's error to the winner's, on 4 systems. In H₂ the interpolatory model wins every time, by 0.8%, 1.3%, 0.4%, 1.9% — IRKA satisfies the first-order conditions for that norm and balanced truncation does not. In H∞ balanced truncation wins every time, by 16%, 23%, 12%, 25%, which is the norm its bound is stated in. Both orderings hold on every system, so this is not a coin flip; and the H₂ margin is under two per cent, so a method costing r solves lands within two per cent of one costing two Lyapunov solves in the norm the cheap one optimises. The choice in this field is about cost.
two-norms is one function in lib/figures/ratkrylov.js —
matching the function where you choose — interpolation points, the basis they need, and the fixed point. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
Ratios of the loser's error to the winner's, on 4 systems. In H₂ the interpolatory model wins every time, by 0.8%, 1.3%, 0.4%, 1.9% — IRKA satisfies the first-order conditions for that norm and balanced truncation does not. In H∞ balanced truncation wins every time, by 16%, 23%, 12%, 25%, which is the norm its bound is stated in. Both orderings hold on every system, so this is not a coin flip; and the H₂ margin is under two per cent, so a method costing r solves lands within two per cent of one costing two Lyapunov solves in the norm the cheap one optimises. The choice in this field is about cost.
which: "h2"
The arguments are the ones Bracketing an error nobody can measure passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Ratios of the loser's error to the winner's, on 4 systems. In H₂ the interpolatory model wins every time, by 0.8%, 1.3%, 0.4%, 1.9% — IRKA satisfies the first-order conditions for that norm and balanced truncation does not. In H∞ balanced truncation wins every time, by 16%, 23%, 12%, 25%, which is the norm its bound is stated in. Both orderings hold on every system, so this is not a coin flip; and the H₂ margin is under two per cent, so a method costing r solves lands within two per cent of one costing two Lyapunov solves in the norm the cheap one optimises. The choice in this field is about cost.
which: "both"
The arguments are the ones Interpolating at the model’s own poles passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Ratios of the loser's error to the winner's, on 4 systems. In H₂ the interpolatory model wins every time, by 0.8%, 1.3%, 0.4%, 1.9% — IRKA satisfies the first-order conditions for that norm and balanced truncation does not. In H∞ balanced truncation wins every time, by 16%, 23%, 12%, 25%, which is the norm its bound is stated in. Both orderings hold on every system, so this is not a coin flip; and the H₂ margin is under two per cent, so a method costing r solves lands within two per cent of one costing two Lyapunov solves in the norm the cheap one optimises. The choice in this field is about cost.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
16 distinct claims across 3 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
a degree the state dimension can carry
a grid fine enough to have modes and coarse enough to draw
a Lyapunov equation with a solution
a norm this figure has been drawn for
a projection that exists
a reduction that removes something and keeps something
a truncation above the numerical rank of the Gramians
an actuator and a sensor on the grid
an interpolation point that adds a direction
and balanced truncation wins H∞
IRKA wins H₂ on every system
Jacobi needs a symmetric matrix
LU is for square matrices
matmul shapes agree
the projection is biorthogonal as normalised
with an H₂ margin under five per cent
Against the rule
It calls a factoriser without drawing a factorisation
(balancedTruncation, reductionError, irka, h2Error),
so the rule is written down as not applying, with the reason:
the bars are ratios of two measured errors, each a norm of a function obtained through a Lyapunov solve; nothing factorised is displayed and the comparison is the whole figure
The exemption list is the interesting half of the rule rather than an escape hatch — it is
where a decision about a figure had to be argued in one line. residualcheck
refuses an exemption that is not doing work, and rejected ten of the fifteen written for the
expansion's figures on exactly that ground: a figure whose vertical axis is a residual
satisfies the rule by construction, and touching a factoriser does not by itself require an
entry.
Across the library: the rule bites on 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
Bracketing an error nobody can measure
The error of a low-rank Gramian factor is the one quantity a caller cannot compute, because computing it needs the Gramian the factor exists to avoid forming. Two numbers that can be computed sit either side of it — a rational factor known before the run, and a residual known after — and they stay a factor of four apart across a fourfold change of size.
Reduction, and what a model is forInterpolating 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.