cluster-thinning
At its defaults it draws the preconditioned spectrum on four grids (ρ = 0.9). Every eigenvalue of C⁻¹A, drawn as a point above the grid it belongs to, on a logarithmic vertical axis. The shaded band is within half a unit of one. The number of eigenvalues inside it goes 9, 11, 13, 17 while the number of unknowns goes 16, 36, 64, 100 — so the share clustered falls from 56% to 17%.
cluster-thinning is one function in lib/figures/bttb.js —
block toeplitz — the second dimension, where the cluster thins. 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.
Every eigenvalue of C⁻¹A, drawn as a point above the grid it belongs to, on a logarithmic vertical axis. The shaded band is within half a unit of one. The number of eigenvalues inside it goes 9, 11, 13, 17 while the number of unknowns goes 16, 36, 64, 100 — so the share clustered falls from 56% to 17%.
rho: 0.9
The arguments are the ones A limit the matrix never reaches 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.
Every eigenvalue of C⁻¹A, drawn as a point above the grid it belongs to, on a logarithmic vertical axis. The shaded band is within half a unit of one. The number of eigenvalues inside it goes 9, 11, 13, 17 while the number of unknowns goes 16, 36, 64, 100 — so the share clustered falls from 56% to 17%.
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.
13 distinct claims across 2 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.
so the share inside falls at m = 6 — asserted 3 times
the cluster grows at m = 6 — asserted 3 times
a correlation the family is conditioned at
a grid the dense eigensolve is affordable on
and grows like the side rather than the area
Jacobi needs a symmetric matrix
matmul shapes agree
the similarity transform is symmetric
the two-dimensional operator is positive definite
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 limit the matrix never reaches
Szegő's theorem gives a Toeplitz family's condition number in closed form — ((1+ρ)/(1−ρ))², which is 81 at ρ = 0.8. The 8×8 section reaches 52% of it, the 128×128 reaches 98.9%, and none of them ever arrives. A statement about a family is not a statement about the matrix in front of you.
Structure, and the solver that cannot see itA preconditioner that changes sign
Strang's circulant preconditioner takes Toeplitz conjugate gradients from 179 steps to 10 at n = 256. At n = 64 on the same family it takes 66 steps to 109 — worse than doing nothing. Between those rows the preconditioner's smallest eigenvalue crosses zero, and nothing in the published account of the method mentions that it can be negative.
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.
Structure, and the solver that cannot see itTwo dimensions, and the cluster that thins
The same kernel, the same averaging, the same transform — applied along two axes instead of one. In one dimension the preconditioned step count is 7, 10, 10, 10; on square grids with the same unknown counts it is 10, 18, 20, 21, and the share of the spectrum near one falls from 56% to 17%.