Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

gramian-decay

One function in the lowrank library, called 16 times across 9 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 7 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

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 Z_k², where Z_k 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.

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.

Why a Gramian can be truncated at all: a rational approximation problemλₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 30-state model, against Z_k², where Z_k 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.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

λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 30-state model, against Z_k², where Z_k 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.

n: 30

The arguments are the ones An error committed before the arithmetic passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Why a Gramian can be truncated at all: a rational approximation problemλₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 30-state model, against Z_k², where Z_k 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.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

λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 30-state model, against Z_k², where Z_k 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.

n: 40

The arguments are the ones Where to put the poles of a rational function passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Why a Gramian can be truncated at all: a rational approximation problemλₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 40-state model, against Z_k², where Z_k 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 1.11·10⁻⁷ 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 6714, whose ratio is 680.6 — 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.1234567891010⁻¹⁹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹kλₖ₊₁ ÷ λ₁, and the boundZ_k²the Gramianpredicted from two numbersstates40κ of the spectrum681λ11 ÷ λ₁1.1·10⁻⁷the bound there0.001the cliff everything rests onand the reason for it

λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 40-state model, against Z_k², where Z_k 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 1.11·10⁻⁷ 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 6714, whose ratio is 680.6 — 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.

n: 10

The arguments are the ones Why a Gramian can be truncated at all passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Why a Gramian can be truncated at all: a rational approximation problemλₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 10-state model, against Z_k², where Z_k 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.12345678910⁻¹⁹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹kλₖ₊₁ ÷ λ₁, and the boundZ_k²the Gramianpredicted from two numbersstates10κ of the spectrum48λ10 ÷ λ₁8.7·10⁻¹⁴the bound there2.6·10⁻⁵the cliff everything rests onand the reason for it

λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 10-state model, against Z_k², where Z_k 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.

n: 16

The arguments are the ones Why a Gramian can be truncated at all passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Why a Gramian can be truncated at all: a rational approximation problemλₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 16-state model, against Z_k², where Z_k 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.1234567891010⁻¹⁹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹kλₖ₊₁ ÷ λ₁, and the boundZ_k²the Gramianpredicted from two numbersstates16κ of the spectrum116λ11 ÷ λ₁1.2·10⁻¹¹the bound there6.7·10⁻⁵the cliff everything rests onand the reason for it

λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 16-state model, against Z_k², where Z_k 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.

n: 22

The arguments are the ones Why a Gramian can be truncated at all passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

Why a Gramian can be truncated at all: a rational approximation problemλₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 22-state model, against Z_k², where Z_k 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.1234567891010⁻¹⁹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹kλₖ₊₁ ÷ λ₁, and the boundZ_k²the Gramianpredicted from two numbersstates22κ of the spectrum214λ11 ÷ λ₁1.2·10⁻⁹the bound there2.1·10⁻⁴the cliff everything rests onand the reason for it

λₖ₊₁(P)/λ₁(P) for the controllability Gramian of a 22-state model, against Z_k², where Z_k 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.

What it checked while drawing

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions 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.

a Gramian the closed form can be checked against

a grid fine enough to have modes and coarse enough to draw

a system with a known eigenbasis

an actuator and a sensor on the grid

every ratio is under the bound

Jacobi needs a symmetric matrix

matmul shapes agree

Against the rule

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 192 of 346 generators — 174 print a residual and 18 are exempt with a published reason; 154 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.

The eigenvalue problem that is not linear

An error committed before the arithmetic

Before a nonlinear eigenvalue problem is solved, somebody says where they think the eigenvalues are. That sentence sets the accuracy of everything that follows by five orders, costs nothing to say, and cannot be revised once the approximation built on it is in hand.

 Reduction, and what a model is for

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.

 Reduction, and what a model is for

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.

 Reduction, and what a model is for

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.

 Reduction, and what a model is for

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.

 Two errors, and whose fault they are

Three errors and one number

This site's identity has two factors and a division of blame between them. Two fields have now added a third party and a fourth, and only one of the four is a property of anything — the others are decisions, made before the arithmetic, reported by nothing.

 The eigenvalue problem that is not linear

Two approximants and one matrix size

A polynomial approximant linearises to nd rows and a rational one to n(m+1), so the fair contest fixes the matrix and varies the basis. On an easy target set the two are indistinguishable and the ordering flips with the noise; on one that reaches a branch point the rational pulls away by two orders.

 Reduction, and what a model is for

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.

 Reduction, and what a model is for

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.

The whole library · All essays · What must fail