Iterative refinement of the inverse-and-multiply solve at κ = 10^14

At its defaults it draws iterative refinement of the inverse-and-multiply solve at κ = 10^14. The backward error starts at 4.5·10⁻⁵ and falls by about κu a step — 8.1·10⁻⁹, 5.3·10⁻¹², 3.3·10⁻¹⁵, 3.1·10⁻¹⁷, 2.8·10⁻¹⁷ — reaching the LU route's 2.2·10⁻¹⁷ after 5 corrections, each costing 2n² flops against the 2n³ the inversion cost. The forward error, drawn above, does not improve: 0.015 to 7.1·10⁻⁴, against the LU route's 2.8·10⁻⁴. What refinement at working precision buys is stability, and the accuracy floor belongs to the problem.

refinement-trail is one function in lib/figures/inverse.js — the inverse — an object almost nobody needs, and the backward error of forming it. 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.

The backward error starts at 4.5·10⁻⁵ and falls by about κu a step — 8.1·10⁻⁹, 5.3·10⁻¹², 3.3·10⁻¹⁵, 3.1·10⁻¹⁷, 2.8·10⁻¹⁷ — reaching the LU route's 2.2·10⁻¹⁷ after 5 corrections, each costing 2n² flops against the 2n³ the inversion cost. The forward error, drawn above, does not improve: 0.015 to 7.1·10⁻⁴, against the LU route's 2.8·10⁻⁴. What refinement at working precision buys is stability, and the accuracy floor belongs to the problem.

The backward error starts at 4.5·10⁻⁵ and falls by about κu a step — 8.1·10⁻⁹, 5.3·10⁻¹², 3.3·10⁻¹⁵, 3.1·10⁻¹⁷, 2.8·10⁻¹⁷ — reaching the LU route's 2.2·10⁻¹⁷ after 5 corrections, each costing 2n² flops against the 2n³ the inversion cost. The forward error, drawn above, does not improve: 0.015 to 7.1·10⁻⁴, against the LU route's 2.8·10⁻⁴. What refinement at working precision buys is stability, and the accuracy floor belongs to the problem.

The backward error starts at 1.9·10⁻¹² and falls by about κu a step — 2·10⁻¹⁷, 2.4·10⁻¹⁷, 2.3·10⁻¹⁷, 2·10⁻¹⁷, 1.9·10⁻¹⁷ — reaching the LU route's 3.3·10⁻¹⁷ after 5 corrections, each costing 2n² flops against the 2n³ the inversion cost. The forward error, drawn above, does not improve: 2.6·10⁻¹¹ to 9.2·10⁻¹², against the LU route's 1.2·10⁻¹¹. What refinement at working precision buys is stability, and the accuracy floor belongs to the problem.

The backward error starts at 6.4·10⁻⁹ and falls by about κu a step — 1.5·10⁻¹⁶, 2.2·10⁻¹⁷, 2.4·10⁻¹⁷, 2.4·10⁻¹⁷, 2·10⁻¹⁷ — reaching the LU route's 2.9·10⁻¹⁷ after 5 corrections, each costing 2n² flops against the 2n³ the inversion cost. The forward error, drawn above, does not improve: 1.1·10⁻⁶ to 4.9·10⁻⁸, against the LU route's 2·10⁻⁸. What refinement at working precision buys is stability, and the accuracy floor belongs to the problem.

The backward error starts at 5.7·10⁻⁶ and falls by about κu a step — 7.3·10⁻¹¹, 1.3·10⁻¹⁵, 2.2·10⁻¹⁷, 1.9·10⁻¹⁷, 2.2·10⁻¹⁷ — reaching the LU route's 2.5·10⁻¹⁷ after 5 corrections, each costing 2n² flops against the 2n³ the inversion cost. The forward error, drawn above, does not improve: 0.001 to 2.7·10⁻⁵, against the LU route's 6.3·10⁻⁵. What refinement at working precision buys is stability, and the accuracy floor belongs to the problem.

The backward error starts at 2.4·10⁻⁷ and falls by about κu a step — 2.4·10⁻¹², 2.6·10⁻¹⁷, 1.1·10⁻¹⁷, 1.2·10⁻¹⁷, 5.4·10⁻¹⁸ — reaching the LU route's 5.2·10⁻¹⁸ after 5 corrections, each costing 2n² flops against the 2n³ the inversion cost. The forward error, drawn above, does not improve: 3.4·10⁻⁵ to 3.7·10⁻⁶, against the LU route's 3.7·10⁻⁶. What refinement at working precision buys is stability, and the accuracy floor belongs to the problem.

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.

20 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.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

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.