Both forms of an interior-point step: two condition numbers that climb together and two errors that do not

At its defaults it draws both forms of an interior-point step: two condition numbers that climb together and two errors that do not. One Newton step of an interior-point method for a quadratic programme with 8 unknowns and 6 constraints, 3 of them active, written twice. The augmented form is [[H, Cᵀ], [C, −D⁻¹]] and the condensed form is H + CᵀDC, and they have the same solution. As the barrier parameter μ falls, the diagonal D separates — z/s runs to 1/μ on the active constraints and to μ on the inactive ones — so both condition numbers climb: 29.04 to 3.044·10¹⁵ for the augmented form and 155.8 to 2.403·10¹⁶ for the condensed one, within a factor of 7.9 of each other. The two forward errors, both measured against the exact rational solution of the system the machine is holding, do not follow: the condensed form's rises to 0.3098 and the augmented form's stays at 1.035·10⁻¹⁵ — fifteen correct digits at a condition number of 3.04·10¹⁵.

barrier-error is one function in lib/figures/barrier.js — the barrier — a condition number sent to infinity on purpose, and the number that describes the error. 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.

One Newton step of an interior-point method for a quadratic programme with 8 unknowns and 6 constraints, 3 of them active, written twice. The augmented form is [[H, Cᵀ], [C, −D⁻¹]] and the condensed form is H + CᵀDC, and they have the same solution. As the barrier parameter μ falls, the diagonal D separates — z/s runs to 1/μ on the active constraints and to μ on the inactive ones — so both condition numbers climb: 29.04 to 3.044·10¹⁵ for the augmented form and 155.8 to 2.403·10¹⁶ for the condensed one, within a factor of 7.9 of each other. The two forward errors, both measured against the exact rational solution of the system the machine is holding, do not follow: the condensed form's rises to 0.3098 and the augmented form's stays at 1.035·10⁻¹⁵ — fifteen correct digits at a condition number of 3.04·10¹⁵.

One Newton step of an interior-point method for a quadratic programme with 8 unknowns and 6 constraints, 3 of them active, written twice. The augmented form is [[H, Cᵀ], [C, −D⁻¹]] and the condensed form is H + CᵀDC, and they have the same solution. As the barrier parameter μ falls, the diagonal D separates — z/s runs to 1/μ on the active constraints and to μ on the inactive ones — so both condition numbers climb: 29.04 to 3.044·10¹⁵ for the augmented form and 155.8 to 2.403·10¹⁶ for the condensed one, within a factor of 7.9 of each other. The two forward errors, both measured against the exact rational solution of the system the machine is holding, do not follow: the condensed form's rises to 0.3098 and the augmented form's stays at 1.035·10⁻¹⁵ — fifteen correct digits at a condition number of 3.04·10¹⁵.

One Newton step of an interior-point method for a quadratic programme with 8 unknowns and 6 constraints, 4 of them active, written twice. The augmented form is [[H, Cᵀ], [C, −D⁻¹]] and the condensed form is H + CᵀDC, and they have the same solution. As the barrier parameter μ falls, the diagonal D separates — z/s runs to 1/μ on the active constraints and to μ on the inactive ones — so both condition numbers climb: 31.22 to 2.193·10¹⁵ for the augmented form and 102.3 to 2.794·10¹⁶ for the condensed one, within a factor of 12.7 of each other. The two forward errors, both measured against the exact rational solution of the system the machine is holding, do not follow: the condensed form's rises to 0.06652 and the augmented form's stays at 3.49·10⁻¹⁵ — fifteen correct digits at a condition number of 2.19·10¹⁵.

One Newton step of an interior-point method for a quadratic programme with 8 unknowns and 6 constraints, 1 of them active, written twice. The augmented form is [[H, Cᵀ], [C, −D⁻¹]] and the condensed form is H + CᵀDC, and they have the same solution. As the barrier parameter μ falls, the diagonal D separates — z/s runs to 1/μ on the active constraints and to μ on the inactive ones — so both condition numbers climb: 28.23 to 1.439·10¹⁶ for the augmented form and 162.7 to 4.182·10¹⁷ for the condensed one, within a factor of 29.1 of each other. The two forward errors, both measured against the exact rational solution of the system the machine is holding, do not follow: the condensed form's rises to 1 and the augmented form's stays at 1.352·10⁻¹⁵ — fifteen correct digits at a condition number of 1.44·10¹⁶.

One Newton step of an interior-point method for a quadratic programme with 8 unknowns and 6 constraints, 2 of them active, written twice. The augmented form is [[H, Cᵀ], [C, −D⁻¹]] and the condensed form is H + CᵀDC, and they have the same solution. As the barrier parameter μ falls, the diagonal D separates — z/s runs to 1/μ on the active constraints and to μ on the inactive ones — so both condition numbers climb: 28.34 to 1.401·10¹⁶ for the augmented form and 160.6 to 1.673·10¹⁷ for the condensed one, within a factor of 11.9 of each other. The two forward errors, both measured against the exact rational solution of the system the machine is holding, do not follow: the condensed form's rises to 1 and the augmented form's stays at 5.024·10⁻¹⁵ — fifteen correct digits at a condition number of 1.4·10¹⁶.

One Newton step of an interior-point method for a quadratic programme with 8 unknowns and 6 constraints, 5 of them active, written twice. The augmented form is [[H, Cᵀ], [C, −D⁻¹]] and the condensed form is H + CᵀDC, and they have the same solution. As the barrier parameter μ falls, the diagonal D separates — z/s runs to 1/μ on the active constraints and to μ on the inactive ones — so both condition numbers climb: 31.53 to 2.161·10¹⁵ for the augmented form and 97.8 to 1.482·10¹⁶ for the condensed one, within a factor of 6.86 of each other. The two forward errors, both measured against the exact rational solution of the system the machine is holding, do not follow: the condensed form's rises to 0.2882 and the augmented form's stays at 4.805·10⁻¹⁵ — fifteen correct digits at a condition number of 2.16·10¹⁵.

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.

31 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 barrierSweep, 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.