The transfer function of a 24-state model, by a solve and by a formula

At its defaults it draws the transfer function of a 24-state model, by a solve and by a formula. |H(iω)| for the second-difference operator on 24 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 24 × 24 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 1.05·10⁻¹⁴ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 3.85·10⁻¹⁵ at every frequency. A state dimension of 24 and a function of one variable: the gap between those is what the field is about.

transfer-response is one function in lib/figures/lti.js — a model that is a rational function — the degree that is a rank, by two routes that disagree. 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.

|H(iω)| for the second-difference operator on 24 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 24 × 24 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 1.05·10⁻¹⁴ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 3.85·10⁻¹⁵ at every frequency. A state dimension of 24 and a function of one variable: the gap between those is what the field is about.

|H(iω)| for the second-difference operator on 24 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 24 × 24 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 1.05·10⁻¹⁴ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 3.85·10⁻¹⁵ at every frequency. A state dimension of 24 and a function of one variable: the gap between those is what the field is about.

|H(iω)| for the second-difference operator on 12 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 12 × 12 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 2.96·10⁻¹⁵ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 1.23·10⁻¹⁵ at every frequency. A state dimension of 12 and a function of one variable: the gap between those is what the field is about.

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.

8 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.

It draws a decomposition and prints its residual. It calls transferBySolve, hinfNorm, 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.