rotation-failure
At its defaults it draws smoothed aggregation against the angle of the anisotropy (ε = 0.001). Convergence factor against the rotation angle, with the stencil's largest axis coupling and its diagonal coupling on the same axis. Aligned, the factor is 0.1935. At 45° it is 0.7894, and the reason is beside it: the axis couplings are 0.5005 while the coupling along the direction the anisotropy runs in is 0.2497. The matrix does not contain the anisotropy.
rotation-failure is one function in lib/figures/aggregation.js —
smoothed aggregation — the sweep that is the method, and the angle that defeats it. 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.
Convergence factor against the rotation angle, with the stencil's largest axis coupling and its diagonal coupling on the same axis. Aligned, the factor is 0.1935. At 45° it is 0.7894, and the reason is beside it: the axis couplings are 0.5005 while the coupling along the direction the anisotropy runs in is 0.2497. The matrix does not contain the anisotropy.
eps: 0.001
The arguments are the ones A direction the smoother cannot see 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.
Convergence factor against the rotation angle, with the stencil's largest axis coupling and its diagonal coupling on the same axis. Aligned, the factor is 0.1935. At 45° it is 0.7894, and the reason is beside it: the axis couplings are 0.5005 while the coupling along the direction the anisotropy runs in is 0.2497. The matrix does not contain the anisotropy.
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.
5 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.
a grid the hierarchy has room to coarsen on
an anisotropy strong enough for the direction to matter
and fails at 45°
the method converges on the aligned operator
where the largest coupling is along an axis rather than along the anisotropy
Against the rule
It draws a decomposition and prints its residual. It calls
saSolve,
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 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 direction the smoother cannot see
Give the Laplacian a strong direction and multigrid stops working — from 0.2016 a cycle to 0.9565 — with every component unchanged and the condition number identical to twelve digits. The problem did not get harder. The link between the method's two halves broke.
Iterating, instead of factorisingAggregating what the matrix calls strong
The depth phase measured every method it had on the 45°-rotated anisotropic operator — 0.784, 0.883, 0.844 — and diagnosed the failure as being in the discretisation rather than in the hierarchy. Smoothed aggregation is the standard answer to anisotropy. It returns 0.789.
Iterating, instead of factorisingCoarsening in one direction only
Leave the smoother alone and halve only the strong direction, and the smoothing factor is 0.3340 — identical to line relaxation's, at every anisotropy and every weight, to twelve digits. The convergence factors are then a factor of three apart, and at 45° both repairs fail outright.
Iterating, instead of factorisingSmoothing a whole line at once
Solve every grid line in the strong direction exactly rather than sweeping over it, and the smoothing factor goes from 0.9993 back to 0.3340 — which is the one-dimensional answer, on a problem that is not one-dimensional. The repair replaces one ε in the closed form by a one.