Why a Gramian can be truncated at all: a rational approximation problem

At its defaults it draws why a gramian can be truncated at all: a rational approximation problem. λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 30-state model, against Zₖ², where Zₖ is the smallest a degree-(k,k) rational function with poles on one side can be made on the other. The Gramian's spectrum falls from 1 to 2.48·10⁻⁸ in 10 steps, and the bound falls with it. Nothing about the model enters the bound except the two ends of the spectrum of −A, 9.86 and 3834, whose ratio is 388.8 — so the decay that makes model reduction possible is predictable from two numbers before the Gramian exists. That is the fact this whole field rests on and it is a fact about rational approximation.

gramian-decay is one function in lib/figures/lowrank.js — why a gramian can be truncated — a rational function's poles, and the decay they explain. 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.

λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 30-state model, against Zₖ², where Zₖ is the smallest a degree-(k,k) rational function with poles on one side can be made on the other. The Gramian's spectrum falls from 1 to 2.48·10⁻⁸ in 10 steps, and the bound falls with it. Nothing about the model enters the bound except the two ends of the spectrum of −A, 9.86 and 3834, whose ratio is 388.8 — so the decay that makes model reduction possible is predictable from two numbers before the Gramian exists. That is the fact this whole field rests on and it is a fact about rational approximation.

λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 26-state model, against Zₖ², where Zₖ is the smallest a degree-(k,k) rational function with poles on one side can be made on the other. The Gramian's spectrum falls from 1 to 7.36·10⁻⁹ in 10 steps, and the bound falls with it. Nothing about the model enters the bound except the two ends of the spectrum of −A, 9.86 and 2906, whose ratio is 294.8 — so the decay that makes model reduction possible is predictable from two numbers before the Gramian exists. That is the fact this whole field rests on and it is a fact about rational approximation.

λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 30-state model, against Zₖ², where Zₖ is the smallest a degree-(k,k) rational function with poles on one side can be made on the other. The Gramian's spectrum falls from 1 to 2.48·10⁻⁸ in 10 steps, and the bound falls with it. Nothing about the model enters the bound except the two ends of the spectrum of −A, 9.86 and 3834, whose ratio is 388.8 — so the decay that makes model reduction possible is predictable from two numbers before the Gramian exists. That is the fact this whole field rests on and it is a fact about rational approximation.

λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 10-state model, against Zₖ², where Zₖ is the smallest a degree-(k,k) rational function with poles on one side can be made on the other. The Gramian's spectrum falls from 1 to 8.7·10⁻¹⁴ in 9 steps, and the bound falls with it. Nothing about the model enters the bound except the two ends of the spectrum of −A, 9.8 and 474.2, whose ratio is 48.37 — so the decay that makes model reduction possible is predictable from two numbers before the Gramian exists. That is the fact this whole field rests on and it is a fact about rational approximation.

λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 16-state model, against Zₖ², where Zₖ is the smallest a degree-(k,k) rational function with poles on one side can be made on the other. The Gramian's spectrum falls from 1 to 1.24·10⁻¹¹ in 10 steps, and the bound falls with it. Nothing about the model enters the bound except the two ends of the spectrum of −A, 9.84 and 1146, whose ratio is 116.5 — so the decay that makes model reduction possible is predictable from two numbers before the Gramian exists. That is the fact this whole field rests on and it is a fact about rational approximation.

λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 22-state model, against Zₖ², where Zₖ is the smallest a degree-(k,k) rational function with poles on one side can be made on the other. The Gramian's spectrum falls from 1 to 1.19·10⁻⁹ in 10 steps, and the bound falls with it. Nothing about the model enters the bound except the two ends of the spectrum of −A, 9.85 and 2106, whose ratio is 213.7 — so the decay that makes model reduction possible is predictable from two numbers before the Gramian exists. That is the fact this whole field rests on and it is a fact about rational approximation.

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.

7 distinct claims across 6 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.

It draws a decomposition and prints its residual. It calls gramianSpectrum, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

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.