The same Householder QR on the same rows, in two orders
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.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
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.
rows: 12
The arguments are the ones A constraint is a weight at infinity passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
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.
rows: 30
The arguments are the ones A constraint is a weight at infinity passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
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.
What it checked while drawing
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.
12 distinct claims across 3 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.
τ = 1: and agree while the weight is mild — checked 4 times
a finite double, since an infinity is not a rational
a method this file implements
a problem with more data than unknowns
a problem with more data than unknowns and fewer constraints
a row order this file implements
the optimality conditions have a solution
the two orders separate by six orders or more at the largest weight
τ = 10⁴: and agree while the weight is mild
Against the rule
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 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.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A constraint is a weight at infinity
Stack an equality constraint on top of a least-squares problem with a large weight and the answer approaches the constrained one like 1/τ². The limit is takeable to any accuracy — and how far it can be taken is a property of the solver, not of the problem. One of them stops at the square root of the precision, and one of them does not stop.
Least squares, and the road not to takeThe weight the factor met first
The route to one minus a leverage through the orthogonal factor was said to lose a digit for every decade of the condition number, whatever else it does. Put a weight on one row and it does not. With the heavy row first, the complement keeps every digit at κ(A) = 2.5·10⁹ while both subtractions return nothing. With the same row last it loses digits as the row's scale grows. And two heavy rows that leave κ(A) at 3.1 still lose six digits when the light rows come first. The law was about the order the factor met the rows, and the condition number had been standing in for it.