orthogonality-vs-kappa
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 A reduction that changes the order passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run 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 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.
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 — asserted 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
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 66
of 131 generators —
51 print a residual and
15 are exempt with a published reason;
65 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 reduction that changes the order
A tall-skinny QR computed as a tree of independent block factorisations touches a 512×12 matrix once instead of twelve times, computes a completely different sequence of roundings from the sweep it replaces, and returns ‖AᵀA − RᵀR‖/‖AᵀA‖ = 1.65·10⁻¹⁵ against the sweep's 9.95·10⁻¹⁵. On the same matrix classical Gram–Schmidt returns 4.6·10⁻¹⁰.
Orthogonality, measuredOrthogonal is a number
"Q is orthogonal" is a claim about a measurable quantity, ‖QᵀQ − I‖, and on the eight-by-eight Hilbert matrix two standard algorithms return 10⁻¹⁵ and 1 for it. The one that returns 1 still reconstructs the matrix perfectly, which is why nothing warns you.
Two errors, and whose fault they areThe condition number is an amplifier
κ is usually introduced as a definition and then quoted. It is a measurement: perturb the input by a known amount, look at how much the output moves, and the largest ratio you can find is the number.
Where the flop count stopped predicting the timeThe 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.
Orthogonality, measuredTwo Gram–Schmidts
One argument changes. Classical Gram–Schmidt projects the original column onto each previous direction; modified projects what is left of it. In exact arithmetic the coefficients are identical. In floating point they differ by eight orders of magnitude in the thing that matters.