The augmented step's two condition numbers, and the one the error obeys

At its defaults it draws the augmented step's two condition numbers, and the one the error obeys. κ₂ is a worst case over all perturbations of the same NORM, and a normwise perturbation is allowed to put its whole budget on the smallest entry of a matrix. The componentwise number — Skeel's ‖ |A⁻¹||A||x| ‖ / ‖x‖ — is a worst case over perturbations proportional to the entries, which is what a backward-stable factorisation actually makes. For the augmented interior-point matrix the two are 3.044·10¹³ and 13.25 at μ = 10⁻¹², a ratio of 2.296·10¹², and the measured error is 9.434·10⁻¹⁶ — which is the second number times the unit roundoff and has nothing to do with the first. The componentwise line is still finding its level down to about μ = 10⁻⁴, where the two groups of constraints have not yet separated, and from there it does not move in the fourth digit; where it settles — 13.25 here — is set by how many constraints are active rather than by μ. Eliminating the second block to reach the condensed form destroys the distinction: there the componentwise number is 1.324·10¹⁴, tracking the normwise one, because the large entries have been summed into CᵀDC and are no longer separately identifiable.

barrier-skeel 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.

κ₂ is a worst case over all perturbations of the same NORM, and a normwise perturbation is allowed to put its whole budget on the smallest entry of a matrix. The componentwise number — Skeel's ‖ |A⁻¹||A||x| ‖ / ‖x‖ — is a worst case over perturbations proportional to the entries, which is what a backward-stable factorisation actually makes. For the augmented interior-point matrix the two are 3.044·10¹³ and 13.25 at μ = 10⁻¹², a ratio of 2.296·10¹², and the measured error is 9.434·10⁻¹⁶ — which is the second number times the unit roundoff and has nothing to do with the first. The componentwise line is still finding its level down to about μ = 10⁻⁴, where the two groups of constraints have not yet separated, and from there it does not move in the fourth digit; where it settles — 13.25 here — is set by how many constraints are active rather than by μ. Eliminating the second block to reach the condensed form destroys the distinction: there the componentwise number is 1.324·10¹⁴, tracking the normwise one, because the large entries have been summed into CᵀDC and are no longer separately identifiable.

κ₂ is a worst case over all perturbations of the same NORM, and a normwise perturbation is allowed to put its whole budget on the smallest entry of a matrix. The componentwise number — Skeel's ‖ |A⁻¹||A||x| ‖ / ‖x‖ — is a worst case over perturbations proportional to the entries, which is what a backward-stable factorisation actually makes. For the augmented interior-point matrix the two are 3.044·10¹³ and 13.25 at μ = 10⁻¹², a ratio of 2.296·10¹², and the measured error is 9.434·10⁻¹⁶ — which is the second number times the unit roundoff and has nothing to do with the first. The componentwise line is still finding its level down to about μ = 10⁻⁴, where the two groups of constraints have not yet separated, and from there it does not move in the fourth digit; where it settles — 13.25 here — is set by how many constraints are active rather than by μ. Eliminating the second block to reach the condensed form destroys the distinction: there the componentwise number is 1.324·10¹⁴, tracking the normwise one, because the large entries have been summed into CᵀDC and are no longer separately identifiable.

κ₂ is a worst case over all perturbations of the same NORM, and a normwise perturbation is allowed to put its whole budget on the smallest entry of a matrix. The componentwise number — Skeel's ‖ |A⁻¹||A||x| ‖ / ‖x‖ — is a worst case over perturbations proportional to the entries, which is what a backward-stable factorisation actually makes. For the augmented interior-point matrix the two are 2.52·10¹³ and 11.19 at μ = 10⁻¹², a ratio of 2.252·10¹², and the measured error is 6.091·10⁻¹⁶ — which is the second number times the unit roundoff and has nothing to do with the first. The componentwise line is still finding its level down to about μ = 10⁻⁴, where the two groups of constraints have not yet separated, and from there it does not move in the fourth digit; where it settles — 11.19 here — is set by how many constraints are active rather than by μ. Eliminating the second block to reach the condensed form destroys the distinction: there the componentwise number is 2.759·10¹⁴, tracking the normwise one, because the large entries have been summed into CᵀDC and are no longer separately identifiable.

κ₂ is a worst case over all perturbations of the same NORM, and a normwise perturbation is allowed to put its whole budget on the smallest entry of a matrix. The componentwise number — Skeel's ‖ |A⁻¹||A||x| ‖ / ‖x‖ — is a worst case over perturbations proportional to the entries, which is what a backward-stable factorisation actually makes. For the augmented interior-point matrix the two are 1.063·10¹³ and 11.09 at μ = 10⁻¹², a ratio of 9.585·10¹¹, and the measured error is 8.739·10⁻¹⁶ — which is the second number times the unit roundoff and has nothing to do with the first. The componentwise line is still finding its level down to about μ = 10⁻⁴, where the two groups of constraints have not yet separated, and from there it does not move in the fourth digit; where it settles — 11.09 here — is set by how many constraints are active rather than by μ. Eliminating the second block to reach the condensed form destroys the distinction: there the componentwise number is 1.803·10¹⁴, tracking the normwise one, because the large entries have been summed into CᵀDC and are no longer separately identifiable.

κ₂ is a worst case over all perturbations of the same NORM, and a normwise perturbation is allowed to put its whole budget on the smallest entry of a matrix. The componentwise number — Skeel's ‖ |A⁻¹||A||x| ‖ / ‖x‖ — is a worst case over perturbations proportional to the entries, which is what a backward-stable factorisation actually makes. For the augmented interior-point matrix the two are 1.822·10¹³ and 15.57 at μ = 10⁻¹², a ratio of 1.17·10¹², and the measured error is 6.124·10⁻¹⁶ — which is the second number times the unit roundoff and has nothing to do with the first. The componentwise line is still finding its level down to about μ = 10⁻⁴, where the two groups of constraints have not yet separated, and from there it does not move in the fourth digit; where it settles — 15.57 here — is set by how many constraints are active rather than by μ. Eliminating the second block to reach the condensed form destroys the distinction: there the componentwise number is 4.19·10¹⁴, tracking the normwise one, because the large entries have been summed into CᵀDC and are no longer separately identifiable.

κ₂ is a worst case over all perturbations of the same NORM, and a normwise perturbation is allowed to put its whole budget on the smallest entry of a matrix. The componentwise number — Skeel's ‖ |A⁻¹||A||x| ‖ / ‖x‖ — is a worst case over perturbations proportional to the entries, which is what a backward-stable factorisation actually makes. For the augmented interior-point matrix the two are 5.37·10¹² and 11.34 at μ = 10⁻¹², a ratio of 4.737·10¹¹, and the measured error is 4.243·10⁻¹⁶ — which is the second number times the unit roundoff and has nothing to do with the first. The componentwise line is still finding its level down to about μ = 10⁻⁴, where the two groups of constraints have not yet separated, and from there it does not move in the fourth digit; where it settles — 11.34 here — is set by how many constraints are active rather than by μ. Eliminating the second block to reach the condensed form destroys the distinction: there the componentwise number is 2.414·10¹³, tracking the normwise one, because the large entries have been summed into CᵀDC and are no longer separately identifiable.

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.

11 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.