Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

hankel-values

One function in the gramian library, called 15 times across 7 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 10 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 hankel singular values, and the half of them a product cannot see. σₖ divided by σ₁, for a 20-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 9.67·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 1.8·10³⁶ against 1.34·10¹⁸. Every σ below the line is a term in the error bound, so this is not an academic loss.

hankel-values is one function in lib/figures/gramian.js — the bound known before the computation — hankel singular values, and the product that halves them. 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.

Hankel singular values, and the half of them a product cannot seeσₖ divided by σ₁, for a 20-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 9.67·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 1.8·10³⁶ against 1.34·10¹⁸. Every σ below the line is a term in the error bound, so this is not an academic loss.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

σₖ divided by σ₁, for a 20-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 9.67·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 1.8·10³⁶ against 1.34·10¹⁸. Every σ below the line is a term in the error bound, so this is not an academic loss.

n: 20

The arguments are the ones Exact at the points that were named 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.

Hankel singular values, and the half of them a product cannot seeσₖ divided by σ₁, for a 20-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 9.67·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 1.8·10³⁶ against 1.34·10¹⁸. Every σ below the line is a term in the error bound, so this is not an academic loss.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

σₖ divided by σ₁, for a 20-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 9.67·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 1.8·10³⁶ against 1.34·10¹⁸. Every σ below the line is a term in the error bound, so this is not an academic loss.

n: 14

The arguments are the ones The bound that is known in advance 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.

Hankel singular values, and the half of them a product cannot seeσₖ divided by σ₁, for a 14-state model of McMillan degree 10. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 5 values and then stops, flattening at 9.19·10⁻⁹. The dashed line is σ₁√u = 2.23·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 2.91·10³⁴ against 1.71·10¹⁷. Every σ below the line is a term in the error bound, so this is not an academic loss.13579111310⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1index kσₖ ÷ σ₁σ₁√utwo factorstheir productthe normal equations, againroutes agree to k =5product floors at9.2·10⁻⁹σ₁√u2.2·10⁻⁹κ(P)κ(Q)2.9·10³⁴√ of it1.7·10¹⁷do not form the productthe σ below the line are the bound

σₖ divided by σ₁, for a 14-state model of McMillan degree 10. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 5 values and then stops, flattening at 9.19·10⁻⁹. The dashed line is σ₁√u = 2.23·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 2.91·10³⁴ against 1.71·10¹⁷. Every σ below the line is a term in the error bound, so this is not an academic loss.

n: 34

The arguments are the ones The bound that is known in advance 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.

Hankel singular values, and the half of them a product cannot seeσₖ divided by σ₁, for a 34-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 8.06·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 1.76·10³⁸ against 1.33·10¹⁹. Every σ below the line is a term in the error bound, so this is not an academic loss.1357911131510⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1index kσₖ ÷ σ₁σ₁√utwo factorstheir productthe normal equations, againroutes agree to k =6product floors at8.1·10⁻¹⁰σ₁√u2.3·10⁻⁹κ(P)κ(Q)1.8·10³⁸√ of it1.3·10¹⁹do not form the productthe σ below the line are the bound

σₖ divided by σ₁, for a 34-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 8.06·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 1.76·10³⁸ against 1.33·10¹⁹. Every σ below the line is a term in the error bound, so this is not an academic loss.

n: 26

The arguments are the ones The bound that is known in advance 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.

Hankel singular values, and the half of them a product cannot seeσₖ divided by σ₁, for a 26-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 8.07·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 1.58·10³⁸ against 1.26·10¹⁹. Every σ below the line is a term in the error bound, so this is not an academic loss.1357911131510⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1index kσₖ ÷ σ₁σ₁√utwo factorstheir productthe normal equations, againroutes agree to k =6product floors at8.1·10⁻¹⁰σ₁√u2.3·10⁻⁹κ(P)κ(Q)1.6·10³⁸√ of it1.3·10¹⁹do not form the productthe σ below the line are the bound

σₖ divided by σ₁, for a 26-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 8.07·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 1.58·10³⁸ against 1.26·10¹⁹. Every σ below the line is a term in the error bound, so this is not an academic loss.

n: 10

The arguments are the ones The product nobody had to form 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.

Hankel singular values, and the half of them a product cannot seeσₖ divided by σ₁, for a 10-state model of McMillan degree 6. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 5 values and then stops, flattening at 1.45·10⁻⁹. The dashed line is σ₁√u = 2.39·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 2.01·10³⁵ against 4.49·10¹⁷. Every σ below the line is a term in the error bound, so this is not an academic loss.1357910⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1index kσₖ ÷ σ₁σ₁√utwo factorstheir productthe normal equations, againroutes agree to k =5product floors at1.4·10⁻⁹σ₁√u2.4·10⁻⁹κ(P)κ(Q)2·10³⁵√ of it4.5·10¹⁷do not form the productthe σ below the line are the bound

σₖ divided by σ₁, for a 10-state model of McMillan degree 6. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 5 values and then stops, flattening at 1.45·10⁻⁹. The dashed line is σ₁√u = 2.39·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 2.01·10³⁵ against 4.49·10¹⁷. Every σ below the line is a term in the error bound, so this is not an academic loss.

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.

10 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 degree the model can carry

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 state dimension two Lyapunov solves can afford

an actuator and a sensor on the grid

and the route that fails floors where losing half the digits says it should

Jacobi needs a symmetric matrix

matmul shapes agree

the two routes agree and then stop

Against the rule

It draws a decomposition and prints its residual. It calls hankelSingularValues, 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.

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.

 The eigenvalue problem that is not linear

The conditioning that rises with the ceiling

Higher moments multiply a contour method's ceiling by K and grade its block Hankel over ρ to the 2K, so the two knobs are the same knob. One division per quadrature point separates them, and the measurement of what it is worth grows from twenty to twenty thousand.

 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.

 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.

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