weighting-order

At its defaults it draws the same householder qr on the same rows, in two orders. One weighted least-squares problem, one factorisation, and the only difference is whether the 2 heavily weighted constraint rows are placed above the 12 data rows or below them. The algebra is identical, the operation count is identical, the reflectors are the same reflectors. Up to τ = 10⁴ the two answers agree. At τ = 10¹⁴ they are 4.8·10⁻¹⁵ and 0.00208 — a factor of 4.324·10¹¹, from the sequence of the rows and nothing else. The reason is that the first reflector is built from the first column, and a column whose large entries are at the bottom has them annihilated last, after the small ones have already been added to them and lost.

weighting-order is one function in lib/figures/lse.js — a constraint as a weight — the limit, the ceiling that belongs to the solver, and the order of the rows. 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.

One weighted least-squares problem, one factorisation, and the only difference is whether the 2 heavily weighted constraint rows are placed above the 12 data rows or below them. The algebra is identical, the operation count is identical, the reflectors are the same reflectors. Up to τ = 10⁴ the two answers agree. At τ = 10¹⁴ they are 4.8·10⁻¹⁵ and 0.00208 — a factor of 4.324·10¹¹, from the sequence of the rows and nothing else. The reason is that the first reflector is built from the first column, and a column whose large entries are at the bottom has them annihilated last, after the small ones have already been added to them and lost.

One weighted least-squares problem, one factorisation, and the only difference is whether the 2 heavily weighted constraint rows are placed above the 12 data rows or below them. The algebra is identical, the operation count is identical, the reflectors are the same reflectors. Up to τ = 10⁴ the two answers agree. At τ = 10¹⁴ they are 4.8·10⁻¹⁵ and 0.00208 — a factor of 4.324·10¹¹, from the sequence of the rows and nothing else. The reason is that the first reflector is built from the first column, and a column whose large entries are at the bottom has them annihilated last, after the small ones have already been added to them and lost.

One weighted least-squares problem, one factorisation, and the only difference is whether the 2 heavily weighted constraint rows are placed above the 30 data rows or below them. The algebra is identical, the operation count is identical, the reflectors are the same reflectors. Up to τ = 10⁴ the two answers agree. At τ = 10¹⁴ they are 8.97·10⁻¹⁶ and 0.00171 — a factor of 1.902·10¹², from the sequence of the rows and nothing else. The reason is that the first reflector is built from the first column, and a column whose large entries are at the bottom has them annihilated last, after the small ones have already been added to them and lost.

One weighted least-squares problem, one factorisation, and the only difference is whether the 2 heavily weighted constraint rows are placed above the 8 data rows or below them. The algebra is identical, the operation count is identical, the reflectors are the same reflectors. Up to τ = 10⁴ the two answers agree. At τ = 10¹⁴ they are 1.13·10⁻¹⁵ and 0.0125 — a factor of 1.11·10¹³, from the sequence of the rows and nothing else. The reason is that the first reflector is built from the first column, and a column whose large entries are at the bottom has them annihilated last, after the small ones have already been added to them and lost.

One weighted least-squares problem, one factorisation, and the only difference is whether the 2 heavily weighted constraint rows are placed above the 20 data rows or below them. The algebra is identical, the operation count is identical, the reflectors are the same reflectors. Up to τ = 10⁴ the two answers agree. At τ = 10¹⁴ they are 6.55·10⁻¹⁶ and 0.00784 — a factor of 1.197·10¹³, from the sequence of the rows and nothing else. The reason is that the first reflector is built from the first column, and a column whose large entries are at the bottom has them annihilated last, after the small ones have already been added to them and lost.

One weighted least-squares problem, one factorisation, and the only difference is whether the 3 heavily weighted constraint rows are placed above the 12 data rows or below them. The algebra is identical, the operation count is identical, the reflectors are the same reflectors. Up to τ = 10⁴ the two answers agree. At τ = 10¹⁴ they are 3.29·10⁻¹⁶ and 0.00146 — a factor of 4.431·10¹², from the sequence of the rows and nothing else. The reason is that the first reflector is built from the first column, and a column whose large entries are at the bottom has them annihilated last, after the small ones have already been added to them and lost.

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions 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.

12 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 weightingSweep, 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 165 of 306 generators — 150 print a residual and 15 are exempt with a published reason; 141 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.