threshold-switch
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 drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run 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: 0.002
The arguments are the ones A threshold between fill and growth 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 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: 0.01
The arguments are the ones Structure and stability stop being separable 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 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: 0.05
The arguments are the ones The algorithm the libraries actually run 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 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 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 5 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
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 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.
Sparsity, and what elimination costsA threshold between fill and growth
One number decides how small a pivot an elimination will accept. At 0.001 the factor holds 172 entries and the matrix grows by 1,330; at 1 it holds 260 and grows by 1.2. The libraries ship 0.1, and the measurement says why.
Sparsity, and what elimination costsStructure and stability stop being separable
The sparsest variable to eliminate on this matrix has a diagonal entry of 10⁻¹². Eliminating it produces the smaller factor, reproduces the matrix to 3.8·10⁻¹⁷ — better than pivoting does — and returns an answer wrong in the fifth digit.
Eigenvalues, singular values, rankThe algorithm the libraries actually run
Factorise, multiply the factors back in the other order, repeat. That description is complete and correct and produces something nobody would use — on a matrix with eigenvalues +1 and −1 it does not converge at all, and the subdiagonal entry does not move by so much as a rounding error.
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.
Iterating, instead of factorisingThe spectrum that predicts nothing
For a symmetric matrix the eigenvalues govern how fast an iteration converges. Drop symmetry and they stop governing anything — there is a matrix whose eigenvalues are as evenly spread as eigenvalues can be, on which GMRES makes no progress at all until the last possible step.
Elimination, and the swapThe swap that is not optional
Run elimination without a row interchange on a matrix that needs one and nothing announces a failure. There is no division by zero, no warning, and an answer of the right shape. It is simply wrong, and how wrong depends on a number you did not look at.