What two lines of scaling are worth: the same quadratic in nine systems of units

At its defaults it draws what two lines of scaling are worth: the same quadratic in nine systems of units. The forward error against the closed form for an overdamped chain of 8 masses, before and after Fan–Lin–Van Dooren scaling. The unscaled curve runs 8.36·10⁻¹⁴ to 0.001258 — every digit gone by the far end — and the scaled one runs 1.34·10⁻¹³ to 9.09·10⁻¹⁴, flat to within a factor of 1.91. The scaling is γ, the square root of ‖K‖ over ‖M‖, and δ, two over the sum of ‖K‖ and γ‖C‖, computed from three norms and nothing else, and the map back is λ = γμ with no rounding in the statement. Flatness is the half that matters: after scaling every stop of the sweep IS the same problem, so there is nothing left for the change of units to do.

scaling-recovery is one function in lib/figures/qepscale.js — the units a polynomial is written in — two lines of scaling, a prediction that is half right, and a format's edge. 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 forward error against the closed form for an overdamped chain of 8 masses, before and after Fan–Lin–Van Dooren scaling. The unscaled curve runs 8.36·10⁻¹⁴ to 0.001258 — every digit gone by the far end — and the scaled one runs 1.34·10⁻¹³ to 9.09·10⁻¹⁴, flat to within a factor of 1.91. The scaling is γ, the square root of ‖K‖ over ‖M‖, and δ, two over the sum of ‖K‖ and γ‖C‖, computed from three norms and nothing else, and the map back is λ = γμ with no rounding in the statement. Flatness is the half that matters: after scaling every stop of the sweep IS the same problem, so there is nothing left for the change of units to do.

The forward error against the closed form for an overdamped chain of 8 masses, before and after Fan–Lin–Van Dooren scaling. The unscaled curve runs 8.36·10⁻¹⁴ to 0.001258 — every digit gone by the far end — and the scaled one runs 1.34·10⁻¹³ to 9.09·10⁻¹⁴, flat to within a factor of 1.91. The scaling is γ, the square root of ‖K‖ over ‖M‖, and δ, two over the sum of ‖K‖ and γ‖C‖, computed from three norms and nothing else, and the map back is λ = γμ with no rounding in the statement. Flatness is the half that matters: after scaling every stop of the sweep IS the same problem, so there is nothing left for the change of units to do.

The forward error against the closed form for an overdamped chain of 4 masses, before and after Fan–Lin–Van Dooren scaling. The unscaled curve runs 5.12·10⁻¹⁵ to 4.331·10⁻⁴ — every digit gone by the far end — and the scaled one runs 9.55·10⁻¹⁵ to 6.65·10⁻¹⁵, flat to within a factor of 3.21. The scaling is γ, the square root of ‖K‖ over ‖M‖, and δ, two over the sum of ‖K‖ and γ‖C‖, computed from three norms and nothing else, and the map back is λ = γμ with no rounding in the statement. Flatness is the half that matters: after scaling every stop of the sweep IS the same problem, so there is nothing left for the change of units to do.

The forward error against the closed form for an overdamped chain of 6 masses, before and after Fan–Lin–Van Dooren scaling. The unscaled curve runs 8.71·10⁻¹⁵ to 0.006738 — every digit gone by the far end — and the scaled one runs 2.54·10⁻¹⁴ to 1.07·10⁻¹⁴, flat to within a factor of 7.25. The scaling is γ, the square root of ‖K‖ over ‖M‖, and δ, two over the sum of ‖K‖ and γ‖C‖, computed from three norms and nothing else, and the map back is λ = γμ with no rounding in the statement. Flatness is the half that matters: after scaling every stop of the sweep IS the same problem, so there is nothing left for the change of units to do.

The forward error against the closed form for an overdamped chain of 10 masses, before and after Fan–Lin–Van Dooren scaling. The unscaled curve runs 2.96·10⁻¹³ to 0.07412 — every digit gone by the far end — and the scaled one runs 7.53·10⁻¹⁴ to 1.28·10⁻¹⁴, flat to within a factor of 16.1. The scaling is γ, the square root of ‖K‖ over ‖M‖, and δ, two over the sum of ‖K‖ and γ‖C‖, computed from three norms and nothing else, and the map back is λ = γμ with no rounding in the statement. Flatness is the half that matters: after scaling every stop of the sweep IS the same problem, so there is nothing left for the change of units to do.

The forward error against the closed form for an overdamped chain of 12 masses, before and after Fan–Lin–Van Dooren scaling. The unscaled curve runs 2.61·10⁻¹⁴ to 0.3958 — every digit gone by the far end — and the scaled one runs 1.15·10⁻¹³ to 4.13·10⁻¹⁴, flat to within a factor of 6.31. The scaling is γ, the square root of ‖K‖ over ‖M‖, and δ, two over the sum of ‖K‖ and γ‖C‖, computed from three norms and nothing else, and the map back is λ = γμ with no rounding in the statement. Flatness is the half that matters: after scaling every stop of the sweep IS the same problem, so there is nothing left for the change of units to do.

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.

17 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.