Strong couplings and the points the matrix kept, ε = 1
At its defaults it draws strong couplings and the points the matrix kept, ε = 1. A grid of points with lines drawn between the pairs the matrix couples strongly, and the points the coarsening kept drawn larger. At small anisotropy only the vertical lines remain and whole rows are kept.
strength-graph 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 grid of points with lines drawn between the pairs the matrix couples strongly, and the points the coarsening kept drawn larger. At small anisotropy only the vertical lines remain and whole rows are kept.
eps: 0.001
The arguments are the ones The coarse grid the matrix chooses passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A grid of points with lines drawn between the pairs the matrix couples strongly, and the points the coarsening kept drawn larger. At small anisotropy only the vertical lines remain and whole rows are kept.
eps: 1
The arguments are the ones The coarse grid the matrix chooses passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A grid of points with lines drawn between the pairs the matrix couples strongly, and the points the coarsening kept drawn larger. At small anisotropy only the vertical lines remain and whole rows are kept.
eps: 0.2
The arguments are the ones The coarse grid the matrix chooses passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A grid of points with lines drawn between the pairs the matrix couples strongly, and the points the coarsening kept drawn larger. At small anisotropy only the vertical lines remain and whole rows are kept.
eps: 0.3
The arguments are the ones The coarse grid the matrix chooses passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A grid of points with lines drawn between the pairs the matrix couples strongly, and the points the coarsening kept drawn larger. At small anisotropy only the vertical lines remain and whole rows are kept.
eps: 0.5
The arguments are the ones The coarse grid the matrix chooses passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A grid of points with lines drawn between the pairs the matrix couples strongly, and the points the coarsening kept drawn larger. At small anisotropy only the vertical lines remain and whole rows are kept.
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.
3 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.
a grid small enough for its couplings to be drawn
and below the threshold the weak one is not strong at all
the strong direction is always strong
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 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.
The 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.
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.