Convergence factor against the strength threshold, ε = 0.01
At its defaults it draws convergence factor against the strength threshold, ε = 0.01. A curve of convergence factor against the strength threshold on a logarithmic axis. It is flat at a high value to the left of a marked position and flat at a low value to the right of it.
threshold-switch 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.
A curve of convergence factor against the strength threshold on a logarithmic axis. It is flat at a high value to the left of a marked position and flat at a low value to the right of it.
eps: 0.005
The arguments are the ones A hierarchy with no grid behind it passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A curve of convergence factor against the strength threshold on a logarithmic axis. It is flat at a high value to the left of a marked position and flat at a low value to the right of it.
eps: 0.25
The arguments are the ones The switch does not know which side is better passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A curve of convergence factor against the strength threshold on a logarithmic axis. It is flat at a high value to the left of a marked position and flat at a low value to the right of it.
eps: 0.02
The arguments are the ones The switch does not know which side is better passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A curve of convergence factor against the strength threshold on a logarithmic axis. It is flat at a high value to the left of a marked position and flat at a low value to the right of it.
eps: 0.15
The arguments are the ones The switch does not know which side is better passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A curve of convergence factor against the strength threshold on a logarithmic axis. It is flat at a high value to the left of a marked position and flat at a low value to the right of it.
eps: 0.3
The arguments are the ones The switch does not know which side is better passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A curve of convergence factor against the strength threshold on a logarithmic axis. It is flat at a high value to the left of a marked position and flat at a low value to the right of 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.
5 distinct claims across 6 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 above it the coarsening is semi-coarsening
and at a mild one full coarsening is perfectly good
at a real anisotropy the semi-coarsened side converges far faster
below ε the coarsening is full
the scan straddles ε
Against the rule
It draws a decomposition and prints its residual. It calls
amgSolve,
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 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 switch does not know which side is better
The strength threshold moves the coarsening from full to semi at θ = ε exactly, at every anisotropy. Which of the two converges faster is a separate question with a separate answer, and it changes sign between ε = 0.33 and ε = 0.34 — where nothing whatever happens to the switch.