Jacobi sweeps on the 6×6 Hilbert matrix
At its defaults it draws jacobi sweeps on the 6×6 hilbert matrix. A plot of the off-diagonal norm against sweep number, falling from about one to ten to the minus seventeen in five sweeps, with the matrix shown at three stages beneath it.
jacobi-convergence is one function in lib/figures/spectra.js —
spectra — sensitivity, the symmetric easy case, and rank as a decision. 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.
A plot of the off-diagonal norm against sweep number, falling from about one to ten to the minus seventeen in five sweeps, with the matrix shown at three stages beneath it.
n: 6
The arguments are the ones Symmetry is worth more than precision passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A plot of the off-diagonal norm against sweep number, falling from about one to ten to the minus seventeen in five sweeps, with the matrix shown at three stages beneath it.
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 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.
swept diagonal 0 is the eigenvalue — checked 6 times
a 6×6 grid leaves a cell wide enough for an entry
in at most eight sweeps
Jacobi needs a symmetric matrix
matmul shapes agree
the off-diagonal norm reaches rounding
the trace is unchanged by the rotations agree
Against the rule
It calls a factoriser without drawing a factorisation
(jacobiEigSym),
so the rule is written down as not applying, with the reason:
plots the off-diagonal norm, which is the convergence measurement
The exemption list is the interesting half of the rule rather than an escape hatch — it is
where a decision about a figure had to be argued in one line. residualcheck
refuses an exemption that is not doing work, and rejected ten of the fifteen written for the
expansion's figures on exactly that ground: a figure whose vertical axis is a residual
satisfies the rule by construction, and touching a factoriser does not by itself require an
entry.
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.
Symmetry is worth more than precision
A symmetric matrix gives up its eigenvalues to full accuracy however ill-conditioned it is. An unsymmetric one can move them by the eighth root of a perturbation, so the rounding involved in merely storing the matrix shifts the spectrum by a hundredth.
Eigenvalues, singular values, rankThe algorithm the libraries actually run
Factorise, multiply the factors back in the other order, repeat. That description is complete and correct and produces something nobody would use — on a matrix with eigenvalues +1 and −1 it does not converge at all, and the subdiagonal entry does not move by so much as a rounding error.