The preconditioned spectrum on four grids (ρ = 0.9)
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: 0.92
The arguments are the ones Four orders of conditioning, and four steps passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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%.
rho: 0.85
The arguments are the ones Four orders of conditioning, and four steps passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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%.
rho: 0.95
The arguments are the ones Four orders of conditioning, and four steps passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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%.
rho: 0.9, grids: [8, 9, 10, 11, 12]
The arguments are the ones Four orders of conditioning, and four steps passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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 13, 15, 17, 19, 21 while the number of unknowns goes 64, 81, 100, 121, 144 — so the share clustered falls from 20% to 15%.
rho: 0.98, grids: [4, 6, 8, 10, 12]
The arguments are the ones Four orders of conditioning, and four steps passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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, 21 while the number of unknowns goes 16, 36, 64, 100, 144 — so the share clustered falls from 56% to 15%.
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.
22 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.
so the share inside falls at m = 6 — checked 6 times
the cluster grows at m = 6 — checked 6 times
a correlation the family is conditioned at
a correlation the sweep draws
a grid the dense eigensolve is affordable on
a p-norm kernel between the separable and the isotropic
a two-dimensional kernel this file builds
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 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.
Four orders of conditioning, and four steps
On a 10×10 grid the two-dimensional kernel's condition number runs from 62 at ρ = 0.5 to 818,561 at ρ = 0.98. The preconditioned step count over the same range runs 18, 21, 21, 22, 21, 19, 18, and the count of eigenvalues the preconditioner actually brings within half a unit of one does not move at all — it is 9, 11, 13, 17 at every correlation the figure will draw.
Structure, and the solver that cannot see itThe staircase a separable kernel builds
A block-circulant preconditioner took 18, 21, 22, 21, 19 and 18 steps on a 10 × 10 Toeplitz-block-Toeplitz system as the correlation rose from 0.5 to 0.98 and the unpreconditioned count rose from 43 to 178 — the parameter that makes the problem hard was the one the solver did not notice. That kernel factorises. The isotropic kernel with the same correlation along each axis does not, and on it the preconditioned count climbs 17, 22, 27, 29, 33, 36, while the preconditioner buys a factor of 1.4 where it bought ten. The reason is the spectrum's shape: a product of two one-dimensional spectra is a staircase of ten treads, and a kernel that is not a product gives a ramp of thirty-eight.
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%.