500 random symmetric orderings of a regularised saddle-point matrix, and what each factorisation reproduces

At its defaults it draws 500 random symmetric orderings of a regularised saddle-point matrix, and what each factorisation reproduces. A quasi-definite matrix — H + δI and −γI on its diagonal, A and its transpose off it, with δ = 10⁻⁶ and γ = 10⁻⁶ — has an LDLᵀ factorisation with diagonal D for every symmetric permutation, so the ordering can be chosen for fill with no numerical veto at all. That is the count: 500 of 500 orderings factorise here, against 69.0 per cent for the same matrix with the zero block left where the problem put it. Each bar counts the orderings whose factorisation left a relative residual ‖PKPᵀ − LDLᵀ‖/‖K‖ in that decade, with the tallest holding 136 of the 500. They span 1.56·10⁻¹⁶ to 1.02·10⁻⁷ — 9 orders — and the growth factor across them runs from 1 to 6.43·10⁵ — which is 0.643/γ, an inverse law in the zero block's perturbation that holds at every γ this figure is drawn at. Existence is not stability, and the theorem says only the first.

ordering-legality is one function in lib/figures/quasidef.js — quasi-definite — every ordering legal, and the fill a symbolic phase can promise exactly. 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 quasi-definite matrix — H + δI and −γI on its diagonal, A and its transpose off it, with δ = 10⁻⁶ and γ = 10⁻⁶ — has an LDLᵀ factorisation with diagonal D for every symmetric permutation, so the ordering can be chosen for fill with no numerical veto at all. That is the count: 500 of 500 orderings factorise here, against 69.0 per cent for the same matrix with the zero block left where the problem put it. Each bar counts the orderings whose factorisation left a relative residual ‖PKPᵀ − LDLᵀ‖/‖K‖ in that decade, with the tallest holding 136 of the 500. They span 1.56·10⁻¹⁶ to 1.02·10⁻⁷ — 9 orders — and the growth factor across them runs from 1 to 6.43·10⁵ — which is 0.643/γ, an inverse law in the zero block's perturbation that holds at every γ this figure is drawn at. Existence is not stability, and the theorem says only the first.

The number of positive pivots in the natural-order factorisation of the saddle-point matrix with H + δI and −10⁻⁸I in its blocks, against δ from 0 to 7, for a constrained minimum (reduced curvatures 0.1 to 5) and a saddle on the same constraint (one reduced curvature −1). The minimum's true count is 10 and it is read correctly at every δ. The saddle's true count is 9; it is read correctly below δ = 1 and as 10, a minimum's, from δ = 1.05 on. The quasi-definite guarantee needs δ past −λmin(H): 4.67 for the minimum and 5.08 for the saddle.

The saddle-point matrix with H + δI and −γI on its diagonal and A and its transpose off it, with δ = 10⁻⁸ and γ = 10⁻⁸, on a Hessian with four negative eigenvalues. H + δI is not positive definite, so the matrix is not quasi-definite and no theorem promises a factorisation under every ordering; 300 of 300 orderings factorise here anyway, against 67.0 per cent with the zero block left at zero. Each bar counts the orderings whose factorisation left a relative residual ‖PKPᵀ − LDLᵀ‖/‖K‖ in that decade, with the tallest holding 84 of the 300. They span 3.19·10⁻¹⁶ to 0.00359 — 13 orders — and the growth factor across them runs from 3.04 to 1.131·10⁷.

The relative error after each step of iterative refinement on the minimum, the residual formed with K and the correction solved with the factorisation of the matrix with H + 5I, on a logarithmic axis. Its pivot signs report 10 positive against a true 10. The error falls by 0.9804 a step between steps 100 and 300; the largest |δ/(μ + δ)| over the reduced curvatures μ predicts 0.9804, drawn dashed. After 300 steps the error is 0.00214.

The relative error after each step of iterative refinement on the saddle, the residual formed with K and the correction solved with the factorisation of the matrix with H + 5I, on a logarithmic axis. Its pivot signs report 10 positive against a true 9. The error grows by 1.2500 a step between steps 50 and 90; the largest |δ/(μ + δ)| over the reduced curvatures μ predicts 1.2500, drawn dashed. After 90 steps the error is 1.36·10⁸.

The relative error after each step of iterative refinement on the saddle, the residual formed with K and the correction solved with the factorisation of the matrix with H + 0.45I, on a logarithmic axis. Its pivot signs report 9 positive against a true 9. The error falls by 0.8182 a step between steps 20 and 60; the largest |δ/(μ + δ)| over the reduced curvatures μ predicts 0.8182, drawn dashed. After 60 steps the error is 6.22·10⁻⁶.

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.

211 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 permutationTrial, 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.