derived-weights
At its defaults it draws the interpolation the matrix derived, on 31 points. Two rows of points, the coarse one above the fine one, joined by lines. Each dropped fine point is joined to two kept ones, and each kept point to itself.
derived-weights is one function in lib/figures/amg.js —
algebraic multigrid — a hierarchy with no grid behind it, and what it 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.
Two rows of points, the coarse one above the fine one, joined by lines. Each dropped fine point is joined to two kept ones, and each kept point to itself.
n: 63
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.
Two rows of points, the coarse one above the fine one, joined by lines. Each dropped fine point is joined to two kept ones, and each kept point to itself.
n: 31
The arguments are the ones The coarse grid the matrix chooses 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.
Two rows of points, the coarse one above the fine one, joined by lines. Each dropped fine point is joined to two kept ones, and each kept point to itself.
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.
an odd grid the coarsening can halve
and a constant interpolates to a constant
and the derived weights are the geometric ½ 1 ½
no two kept points are adjacent
Against the rule
It draws a decomposition and prints its residual. It calls
interpolation,
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 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 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 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.
Iterating, instead of factorisingThe same problem on a coarser grid
Restriction, the coarse operator and interpolation are three matrices with nine distinct entries between them. Two of the three are each other's transpose, and their product with the fine operator is the coarse discretisation exactly — not approximately, entry for entry, at every level.