Backward and forward error against the condition number, on 8 × 8 systems

At its defaults it draws backward and forward error against the condition number, on 8 × 8 systems. A log–log plot over twelve decades of condition number, at 8 × 8, twenty seeds a point. The backward error is flat — median 6.12·10⁻¹⁷ at κ = 10 and 2.4·10⁻¹⁷ at κ = 10¹³, worst 1.45·10⁻¹⁶ anywhere on the sweep — while the forward error climbs from 4.25·10⁻¹⁶ to 5.72·10⁻⁵. At the right-hand end the two are a factor of 2.38·10¹² apart, and nothing about the computation that produced them differs.

residual-vs-error is one function in lib/figures/error.js — error — the backward one, the forward one, and the number between them. 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 log–log plot over twelve decades of condition number, at 8 × 8, twenty seeds a point. The backward error is flat — median 6.12·10⁻¹⁷ at κ = 10 and 2.4·10⁻¹⁷ at κ = 10¹³, worst 1.45·10⁻¹⁶ anywhere on the sweep — while the forward error climbs from 4.25·10⁻¹⁶ to 5.72·10⁻⁵. At the right-hand end the two are a factor of 2.38·10¹² apart, and nothing about the computation that produced them differs.

A log–log plot over twelve decades of condition number, at 30 × 30, twenty seeds a point. The backward error is flat — median 1.3·10⁻¹⁶ at κ = 10 and 5.09·10⁻¹⁷ at κ = 10¹³, worst 2.4·10⁻¹⁶ anywhere on the sweep — while the forward error climbs from 1.01·10⁻¹⁵ to 4.28·10⁻⁵. At the right-hand end the two are a factor of 8.42·10¹¹ apart, and nothing about the computation that produced them differs.

A log–log plot over twelve decades of condition number, at 8 × 8, twenty seeds a point. The backward error is flat — median 6.12·10⁻¹⁷ at κ = 10 and 2.4·10⁻¹⁷ at κ = 10¹³, worst 1.45·10⁻¹⁶ anywhere on the sweep — while the forward error climbs from 4.25·10⁻¹⁶ to 5.72·10⁻⁵. At the right-hand end the two are a factor of 2.38·10¹² apart, and nothing about the computation that produced them differs.

A log–log plot over twelve decades of condition number, at 4 × 4, twenty seeds a point. The backward error is flat — median 3.05·10⁻¹⁷ at κ = 10 and 1.83·10⁻¹⁷ at κ = 10¹³, worst 1.2·10⁻¹⁶ anywhere on the sweep — while the forward error climbs from 2.57·10⁻¹⁶ to 7.26·10⁻⁵. At the right-hand end the two are a factor of 3.96·10¹² apart, and nothing about the computation that produced them differs.

A log–log plot over twelve decades of condition number, at 16 × 16, twenty seeds a point. The backward error is flat — median 8.33·10⁻¹⁷ at κ = 10 and 3.5·10⁻¹⁷ at κ = 10¹³, worst 1.46·10⁻¹⁶ anywhere on the sweep — while the forward error climbs from 7.39·10⁻¹⁶ to 7.41·10⁻⁵. At the right-hand end the two are a factor of 2.12·10¹² apart, and nothing about the computation that produced them differs.

A log–log plot over twelve decades of condition number, at 40 × 40, twenty seeds a point. The backward error is flat — median 1.6·10⁻¹⁶ at κ = 10 and 4.9·10⁻¹⁷ at κ = 10¹³, worst 2.7·10⁻¹⁶ anywhere on the sweep — while the forward error climbs from 1.38·10⁻¹⁵ to 5.37·10⁻⁵. At the right-hand end the two are a factor of 1.1·10¹² apart, and nothing about the computation that produced them differs.

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.

13 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 calls a factoriser without drawing a factorisation (solve), so the rule is written down as not applying, with the reason: plots two errors against κ — no factor of any matrix is shown

The exemption list is the interesting half of the rule rather than an escape hatch — it is where a decision about a figure had to be argued in one line. residualcheck refuses an exemption that is not doing work, and rejected ten of the fifteen written for the expansion's figures on exactly that ground: a figure whose vertical axis is a residual satisfies the rule by construction, and touching a factoriser does not by itself require an entry.

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.