‖QᵀQ − I‖ of the implied Q, against the condition number
At its defaults it draws ‖qᵀq − i‖ of the implied q, against the condition number. Loss of orthogonality against κ, both axes logarithmic, for three factorisations of the same 256×8 matrix. The column sweep and the reduction tree run from 2.7·10⁻¹⁴ to 1.5·10⁻¹⁰. Cholesky QR — the one with a single reduction — runs from 2.3·10⁻¹³ to 1.8·10⁻⁵, a fitted slope of 1.97 against the others' near-flat one.
orthogonality-vs-kappa is one function in lib/figures/parallel.js —
messages — the count words cannot make, and what the fewest of them costs. 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.
Loss of orthogonality against κ, both axes logarithmic, for three factorisations of the same 256×8 matrix. The column sweep and the reduction tree run from 2.7·10⁻¹⁴ to 1.5·10⁻¹⁰. Cholesky QR — the one with a single reduction — runs from 2.3·10⁻¹³ to 1.8·10⁻⁵, a fitted slope of 1.97 against the others' near-flat one.
n: 8
The arguments are the ones The message and the word passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Loss of orthogonality against κ, both axes logarithmic, for three factorisations of the same 256×8 matrix. The column sweep and the reduction tree run from 2.7·10⁻¹⁴ to 1.5·10⁻¹⁰. Cholesky QR — the one with a single reduction — runs from 2.3·10⁻¹³ to 1.8·10⁻⁵, a fitted slope of 1.97 against the others' near-flat one.
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.
11 distinct claims across 2 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 sweep loses at most κu at κ = 100 — checked 2 times
a matrix tall enough for the slices to be tall
and the one-reduction method is always the worse
Cholesky QR's loss of orthogonality grows like κ²
each row block is at least as tall as it is wide
enough condition numbers to fit a slope
matmul shapes agree
the sweep loses at most κu at κ = 10·10⁴
the sweep loses at most κu at κ = 10·10⁵
the sweep loses at most κu at κ = 10⁴
Against the rule
It draws a decomposition and prints its residual. It calls
compare,
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.
The message and the word
Three factorisations of one matrix on sixteen processors: 48 communication rounds, 4, and 4. The words sent are 1,170, 1,170 and 2,160 — so the method with the fewest rounds sends the most words, and the count that separates the three is the one no operation count can see.
Least squares, and the road not to takeThe road that squares the problem
The normal equations are the first method every course teaches and the method no library uses. Forming AᵀA squares the condition number, and below ε = √u it does not degrade — it produces a matrix that is exactly singular, from data that was perfectly usable.