A single Newton step solved to eleven inner tolerances, against how far the resulting point is from the root

At its defaults it draws a single newton step solved to eleven inner tolerances, against how far the resulting point is from the root. The iterate is 0.0372 from a root that is known exactly by construction. The same linear system is solved to relative residuals from 0.3 down to 10⁻¹⁴, at 109 and 1126 conjugate gradient iterations, and the resulting point is 0.005319 and 0.002497 from the root. The curve is flat below about 10⁻³: the linearisation is wrong at second order, so the step cannot land closer than the square of the distance it started at — 0.001383 — however exactly it is computed.

inner-plateau is one function in lib/figures/sequence.js — sequences — a solve inside an outer loop, and the accuracy the loop throws away. 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 iterate is 0.0372 from a root that is known exactly by construction. The same linear system is solved to relative residuals from 0.3 down to 10⁻¹⁴, at 109 and 1126 conjugate gradient iterations, and the resulting point is 0.005319 and 0.002497 from the root. The curve is flat below about 10⁻³: the linearisation is wrong at second order, so the step cannot land closer than the square of the distance it started at — 0.001383 — however exactly it is computed.

Inexact Newton with the adaptive forcing rule on a 200-unknown problem at condition number 10⁵, its inner tolerance floored at c times the outer tolerance times ‖b‖ over ‖F‖. On logarithmic axes against c from 0.01 to 10: the forward error over the unguarded run's, and the inner iterations over its 1109. c = 0.01: error 44×, 983 inner, 10 outer; c = 0.03: error 148×, 932 inner, 10 outer; c = 0.1: error 331×, 897 inner, 10 outer; c = 0.3: error 1677×, 844 inner, 10 outer; c = 0.5: error 2636×, 822 inner, 10 outer; c = 1: error 4979×, 782 inner, 10 outer; c = 3: error 18112×, 774 inner, 79 outer, not converged; c = 10: error 43374×, 733 inner, 79 outer, not converged.

Against the outer tolerance on logarithmic axes: the forward error each run reaches, and its total inner iterations drawn on the same axis as a fraction of its range. With the floor the run reaches 1.97·10⁻¹⁰ in 1206 inner iterations at an outer tolerance of 1e-12; without it, 5.25·10⁻¹³ in 1657. Both runs stop when the outer relative residual is below the tolerance, and both do. The floor saves between 9 and 27 per cent of the inner work and costs up to 2636 times the forward error.

Inexact Newton with the adaptive forcing rule on a 200-unknown problem at condition number 10⁵, its inner tolerance floored at c times the outer tolerance times ‖b‖ over ‖F‖. On logarithmic axes against c from 0.01 to 10: the forward error over the unguarded run's, and the inner iterations over its 1657. c = 0.01: error 11×, 1346 inner, 11 outer; c = 0.03: error 42×, 1294 inner, 11 outer; c = 0.1: error 93×, 1262 inner, 11 outer; c = 0.3: error 219×, 1225 inner, 11 outer; c = 0.5: error 376×, 1206 inner, 11 outer; c = 1: error 1005×, 1168 inner, 11 outer; c = 3: error 2229×, 1179 inner, 79 outer, not converged; c = 10: error 9549×, 1139 inner, 79 outer, not converged.

On logarithmic axes. Each line is one problem, conditioning and nonlinearity as labelled; the outer loop is capped at eighty steps. κ 10⁵, λ 3: 10, 10, 10, 10, 10, 10, 79, 79; κ 10³, λ 3: 8, 8, 8, 8, 8, 8, 22, 79; κ 10⁷, λ 3: 9, 9, 9, 9, 9, 10, 79, 79; κ 10⁵, λ 0.3: 9, 9, 9, 9, 9, 9, 79, 79; κ 10⁵, λ 30: 9, 9, 9, 9, 9, 9, 79, 79 for c = 0.01, 0.03, 0.1, 0.3, 0.5, 1, 3, 10. Every problem converges in its usual count up to c = 1 and stalls at the cap from c = 3 or 10.

Twenty runs: five problems, conditioning 10³ to 10⁷ and nonlinearity 0.3 to 30, at outer tolerances 10⁻⁶ to 10⁻¹². Horizontally, the share of inner iterations the floor saves; vertically, on a logarithmic axis, the forward error with the floor over the error without it. The premium runs from 1.00 to 4.81·10⁴ and the saving from 0.0% to 37.7%, with no relation between them: κ 10⁵, λ 3 at 10⁻⁶, 19.6% and 31.7; κ 10⁵, λ 3 at 10⁻⁸, 25.9% and 2636; κ 10⁵, λ 3 at 10⁻¹⁰, 9.0% and 23.2; κ 10⁵, λ 3 at 10⁻¹², 27.2% and 376; κ 10³, λ 3 at 10⁻⁶, 32.5% and 1221; κ 10³, λ 3 at 10⁻⁸, 11.3% and 12.6; κ 10³, λ 3 at 10⁻¹⁰, 31.7% and 4.81·10⁴; κ 10³, λ 3 at 10⁻¹², 18.9% and 587; κ 10⁷, λ 3 at 10⁻⁶, 37.7% and 1.02; κ 10⁷, λ 3 at 10⁻⁸, 0.0% and 1.00; κ 10⁷, λ 3 at 10⁻¹⁰, 2.4% and 1.74; κ 10⁷, λ 3 at 10⁻¹², 17.5% and 300; κ 10⁵, λ 0.3 at 10⁻⁶, 24.5% and 3.70; κ 10⁵, λ 0.3 at 10⁻⁸, 20.3% and 131; κ 10⁵, λ 0.3 at 10⁻¹⁰, 2.4% and 1.69; κ 10⁵, λ 0.3 at 10⁻¹², 19.8% and 374; κ 10⁵, λ 30 at 10⁻⁶, 28.4% and 374; κ 10⁵, λ 30 at 10⁻⁸, 5.5% and 2.92; κ 10⁵, λ 30 at 10⁻¹⁰, 23.8% and 3935; κ 10⁵, λ 30 at 10⁻¹², 10.5% and 44.4.

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.

28 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.