dimension-steps
At its defaults it draws the same preconditioner in one dimension and in two (ρ = 0.9). Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 7, 10, 10, 10 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 20, 21, climbing, against an unpreconditioned 14, 37, 73, 115. Nothing about the construction changed.
dimension-steps 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.
Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 7, 10, 10, 10 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 20, 21, climbing, against an unpreconditioned 14, 37, 73, 115. Nothing about the construction changed.
rho: 0.9
The arguments are the ones The circulant that cannot be indefinite 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.
Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 7, 10, 10, 10 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 20, 21, climbing, against an unpreconditioned 14, 37, 73, 115. Nothing about the construction changed.
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.
10 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.
the preconditioner does not hurt at 4×4 — asserted 4 times
a correlation the family is conditioned at
a grid the dense two-dimensional solve is affordable on
and helps on at least one of the grids drawn
and the two-dimensional one is not
enough grids to show a trend
the one-dimensional count is flat
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.
The circulant that cannot be indefinite
The previous essay found a preconditioner taking 117 steps against an unpreconditioned 59, because its smallest eigenvalue was −0.173. Average the two diagonals instead of choosing between them and the count is 7, 8, 9, 10, 10 across a factor of sixteen in size.
Iterating, instead of factorisingThe coarse problem is a different problem
In one dimension the Galerkin coarse operator is the coarse discretisation, entry for entry — this site asserted it. In two dimensions a five-point operator produces a nine-point coarse one, so the recursion solves a different discretisation at every level below the first, and converges at 0.20 a cycle regardless.
Iterating, instead of factorisingThe same problem on a coarser grid
Restriction, the coarse operator and interpolation are three matrices with nine distinct entries between them. Two of the three are each other's transpose, and their product with the fine operator is the coarse discretisation exactly — not approximately, entry for entry, at every level.
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%.