aggregate-map
At its defaults it draws the aggregates the matrix chooses at ε = 0.001. A 15×15 grid of unknowns, each tinted by the aggregate that owns it. The operator's couplings are 1.0e-3 along x and 1.000 along y, so the strong direction is y — and the aggregates come out 3.00 points tall on average and 1.00 wide. 75 aggregates cover 225 points, 75 of them opened by the first pass and 15 points handed out by the second.
aggregate-map 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.
A 15×15 grid of unknowns, each tinted by the aggregate that owns it. The operator's couplings are 1.0e-3 along x and 1.000 along y, so the strong direction is y — and the aggregates come out 3.00 points tall on average and 1.00 wide. 75 aggregates cover 225 points, 75 of them opened by the first pass and 15 points handed out by the second.
eps: 0.01
The arguments are the ones A hierarchy with no grid behind it 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.
A 15×15 grid of unknowns, each tinted by the aggregate that owns it. The operator's couplings are 1.0e-2 along x and 1.000 along y, so the strong direction is y — and the aggregates come out 3.00 points tall on average and 1.00 wide. 75 aggregates cover 225 points, 75 of them opened by the first pass and 15 points handed out by the second.
eps: 0.001
The arguments are the ones Aggregating what the matrix calls strong 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.
A 15×15 grid of unknowns, each tinted by the aggregate that owns it. The operator's couplings are 1.0e-3 along x and 1.000 along y, so the strong direction is y — and the aggregates come out 3.00 points tall on average and 1.00 wide. 75 aggregates cover 225 points, 75 of them opened by the first pass and 15 points handed out by the second.
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.
4 distinct claims across 3 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 whose cells are large enough to see
an anisotropy the operator is defined at
and the aggregation coarsens by at least a half
every point is in exactly one aggregate
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 hierarchy with no grid behind it
On a graph Laplacian the algebraic V-cycle converges at 0.199 a cycle, its grid complexity is an unremarkable 3.05, and its operator complexity is 17.7 — one level of forty-one unknowns is entirely dense. The number people quote is the one that does not measure the work.
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 factorisingThe coarse grid the matrix chooses
Given a tridiagonal matrix and no information about a grid, the coarsening keeps every other point and derives the weights ½, 1, ½ — the operators the geometric method was handed. Given the anisotropic operator, it discovers semi-coarsening, in the right direction, without a coordinate.
Sparsity, and what elimination costsThe order decides the memory
Four elimination orderings on one matrix give factors of 1,739, 1,354, 1,413 and 1,026 entries. All four factorisations are exact, all four return the same answer, and the one with the better asymptotics is not the one that wins.