A reduced model that interpolates exactly and has a pole in the right half plane

At its defaults it draws a reduced model that interpolates exactly and has a pole in the right half plane. A convection–diffusion system of 30 states at Péclet number 10, whose state matrix is tridiagonal and not symmetric and whose rightmost eigenvalue is -3.49 — stable. It is reduced to order three by two-sided interpolation at {s, 2s, 4s}, and the horizontal axis is s. The curve is the rightmost pole of the reduced model. Above the zero line the model is unstable: it is at 4 of the 19 placements, reaching 24.1. Every one of those unstable models still matches the full system's transfer function at each of its own three interpolation points, to 1.27·10⁻¹⁴ — the construction did exactly what it promised. The dashed line is balanced truncation of the same system at the same order, whose rightmost pole is -0.837: stable, as it is at every order, because stability is a theorem there rather than an outcome.

projection-stability is one function in lib/figures/redstab.js — a model that interpolates exactly and cannot be run — and the method whose stability is a theorem. 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.

A convection–diffusion system of 30 states at Péclet number 10, whose state matrix is tridiagonal and not symmetric and whose rightmost eigenvalue is -3.49 — stable. It is reduced to order three by two-sided interpolation at {s, 2s, 4s}, and the horizontal axis is s. The curve is the rightmost pole of the reduced model. Above the zero line the model is unstable: it is at 4 of the 19 placements, reaching 24.1. Every one of those unstable models still matches the full system's transfer function at each of its own three interpolation points, to 1.27·10⁻¹⁴ — the construction did exactly what it promised. The dashed line is balanced truncation of the same system at the same order, whose rightmost pole is -0.837: stable, as it is at every order, because stability is a theorem there rather than an outcome.

The 30-state convection–diffusion system at Péclet number 10, written in coordinates x = Dz with D diagonal, whose k-th entry is γ raised to the power tk and whose γ is the square root of the ratio of the two off-diagonals, for t from -3 to 3. The transfer function and the eigenvalues (rightmost -3.49) do not change. Top: the edge of the numerical range on a signed logarithmic axis. Bottom: of nineteen placements, how many one-sided models of orders 1, 2, 3 and 6 are unstable. t -3 (cond D 1.4·10⁶): edge 37.9, unstable 11, 6, 1, 2; t -2 (cond D 1.3·10⁴): edge 19.4, unstable 11, 5, 0, 1; t -1.5 (cond D 1186): edge 12.3, unstable 11, 4, 0, 0; t -1 (cond D 112): edge 6.59, unstable 1, 0, 0, 0; t -0.5 (cond D 11): edge 2.16, unstable 0, 0, 0, 0; t 0 (cond D 1): edge -0.986, unstable 0, 0, 0, 0; t 0.5 (cond D 11): edge -2.86, unstable 0, 0, 0, 0; t 1 (cond D 112): edge -3.49, unstable 0, 0, 0, 0; t 1.5 (cond D 1186): edge -2.86, unstable 0, 0, 0, 0; t 2 (cond D 1.3·10⁴): edge -0.986, unstable 0, 0, 0, 0; t 2.5 (cond D 1.3·10⁵): edge 2.16, unstable 0, 0, 0, 0; t 3 (cond D 1.4·10⁶): edge 6.59, unstable 0, 0, 0, 0.

A 30-state centred-difference convection–diffusion system at Péclet number 10, written in coordinates rescaled by a diagonal matrix whose k-th entry is γ to the power -2k, of condition number 1.25·10⁴. Its rightmost eigenvalue is -3.49 and the edge of its numerical range — the largest eigenvalue of the symmetric part of A — is 19.4. For nineteen placements s from 0.1 to 100, the rightmost pole of a two-sided model of order 3 at {s, 2s, 4s}, a one-sided (Galerkin) model of order 3 at the same shifts, and a one-sided model of order 6 at six shifts from s to 4s, on a signed logarithmic axis. Unstable placements: two-sided 4, one-sided order 3 0, one-sided order 6 1. Worst poles: 24.1, -5.83 and 2.44.

The 30-state convection–diffusion system at Péclet number 40, written in coordinates x = Dz with D diagonal, whose k-th entry is γ raised to the power tk and whose γ is the square root of the ratio of the two off-diagonals, for t from -1.5 to 2. The transfer function and the eigenvalues (rightmost -11.5) do not change. Top: the edge of the numerical range on a signed logarithmic axis. Bottom: of nineteen placements, how many one-sided models of orders 1, 2, 3 and 6 are unstable. t -1.5 (cond D 3.1·10¹⁴): edge 78.9, unstable 11, 11, 9, 2; t -1 (cond D 4.6·10⁹): edge 40.6, unstable 10, 10, 8, 2; t -0.75 (cond D 1.8·10⁷): edge 26.6, unstable 10, 7, 6, 2; t -0.5 (cond D 6.8·10⁴): edge 15.4, unstable 10, 4, 2, 2; t -0.25 (cond D 260): edge 6.58, unstable 10, 0, 0, 3; t 0 (cond D 1): edge -0.247, unstable 0, 0, 0, 0; t 0.5 (cond D 6.8·10⁴): edge -8.81, unstable 0, 0, 0, 0; t 1 (cond D 4.6·10⁹): edge -11.5, unstable 0, 0, 0, 0; t 1.5 (cond D 3.1·10¹⁴): edge -8.81, unstable 0, 0, 0, 0; t 2 (cond D 2.1·10¹⁹): edge -0.247, unstable 0, 0, 0, 0.

For rescalings that skew the convection–diffusion operator further than its original coordinates, the number of nineteen placements at which a one-sided model of order 1 (solid) and order 3 (dashed) is unstable, against the condition number of the rescaling on a logarithmic axis. Pe 10: cond 1.4·10⁶ → 11, 1, cond 1.3·10⁴ → 11, 0, cond 1186 → 11, 0, cond 112 → 1, 0, cond 11 → 0, 0, cond 1 → 0, 0; Pe 25: cond 1.2·10⁸ → 10, 6, cond 2.4·10⁵ → 10, 2, cond 1.1·10⁴ → 11, 0, cond 492 → 10, 0, cond 22 → 0, 0, cond 1 → 0, 0; Pe 40: cond 3.1·10¹⁴ → 11, 9, cond 4.6·10⁹ → 10, 8, cond 1.8·10⁷ → 10, 6, cond 6.8·10⁴ → 10, 2, cond 260 → 10, 0, cond 1 → 0, 0.

A 30-state centred-difference convection–diffusion system at Péclet number 10, written in coordinates rescaled by a diagonal matrix whose k-th entry is γ to the power 1k, of condition number 112. Its rightmost eigenvalue is -3.49 and the edge of its numerical range — the largest eigenvalue of the symmetric part of A — is -3.49. For nineteen placements s from 0.1 to 100, the rightmost pole of a two-sided model of order 3 at {s, 2s, 4s}, a one-sided (Galerkin) model of order 3 at the same shifts, and a one-sided model of order 6 at six shifts from s to 4s, on a signed logarithmic axis. Unstable placements: two-sided 4, one-sided order 3 0, one-sided order 6 0. Worst poles: 24.1, -3.91 and -3.49.

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.

130 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.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

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.