QᵀQ from classical Gram–Schmidt and from Householder on the 8×8 Hilbert matrix
At its defaults it draws qᵀq from classical gram–schmidt and from householder on the 8×8 hilbert matrix. Two eight-by-eight tables of QᵀQ. The upper one has ones on the diagonal and entries as large as one off it; the lower one is the identity to three decimal places everywhere.
qq-printed is one function in lib/figures/ortho.js —
orthogonality — ‖qᵀq − i‖ as a measurement rather than an adjective. 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.
Two eight-by-eight tables of QᵀQ. The upper one has ones on the diagonal and entries as large as one off it; the lower one is the identity to three decimal places everywhere.
n: 8
The arguments are the ones A reflection cannot stop being one passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two eight-by-eight tables of QᵀQ. The upper one has ones on the diagonal and entries as large as one off it; the lower one is the identity to three decimal places everywhere.
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.
6 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.
and both factorisations still reconstruct A
and yet some pair of them is nearly parallel
every classical column is still a unit vector
matmul shapes agree
the orthogonality tolerance sits between the measured noise floor and the smallest real failure
while Householder's columns are perpendicular
Against the rule
It draws a decomposition and prints its residual. It calls
qrHouseholder, gramSchmidt,
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 reflection cannot stop being one
Householder QR holds orthogonality at 10⁻¹⁵ whatever the condition number of the matrix, and Gram–Schmidt does not. The reason is not that it is more careful. It is that its Q is built from unit vectors, and rounding a unit vector gives a different reflection rather than a broken one.
Where the flop count stopped predicting the timeDoing it twice
Cholesky QR squares the condition number — a fitted slope of 1.95 in κ against the Householder sweep's 1.00. Run the identical routine a second time on the Q it returned and the slope is 0.93, the orthogonality is at or below the sweep's at every κ, and the price is one more all-reduce.
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.
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.