The diagonals of T, and what each circulant puts in their place (n = 16, ρ = 0.9)
At its defaults it draws the diagonals of t, and what each circulant puts in their place (n = 16, ρ = 0.9). Entry value against the diagonal index. The Toeplitz matrix's own diagonals fall geometrically as ρ^k. The wrapped circulant follows them to the halfway point and then jumps to the far diagonal, discarding what it passed. The averaged one blends the two throughout, weighted by how many entries each contributes — and the result is nearer to T in the Frobenius norm, 0.2112 against 0.2877.
circulant-construction is one function in lib/figures/chan.js —
two circulants — the one that chooses a diagonal, and the one that averages. 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.
Entry value against the diagonal index. The Toeplitz matrix's own diagonals fall geometrically as ρ^k. The wrapped circulant follows them to the halfway point and then jumps to the far diagonal, discarding what it passed. The averaged one blends the two throughout, weighted by how many entries each contributes — and the result is nearer to T in the Frobenius norm, 0.2112 against 0.2877.
n: 32, rho: 0.9
The arguments are the ones A speedup with a ceiling of its own passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Entry value against the diagonal index. The Toeplitz matrix's own diagonals fall geometrically as ρ^k. The wrapped circulant follows them to the halfway point and then jumps to the far diagonal, discarding what it passed. The averaged one blends the two throughout, weighted by how many entries each contributes — and the result is nearer to T in the Frobenius norm, 0.2765 against 0.3123.
n: 16, rho: 0.9
The arguments are the ones The circulant that cannot be indefinite passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Entry value against the diagonal index. The Toeplitz matrix's own diagonals fall geometrically as ρ^k. The wrapped circulant follows them to the halfway point and then jumps to the far diagonal, discarding what it passed. The averaged one blends the two throughout, weighted by how many entries each contributes — and the result is nearer to T in the Frobenius norm, 0.2112 against 0.2877.
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.
5 distinct claims across 3 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 correlation the entries decay at
a size whose diagonals fit on the axis
and it is the nearer of the two
the averaging formula is the least-squares fit
the Toeplitz corner is one number
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.
A speedup with a ceiling of its own
At ρ = 0.5 the averaged circulant takes 5 conjugate gradient steps at n = 512 against an unpreconditioned 30 — and that 30 is where the unpreconditioned count stops. It reads 29, 28, 30, 30 at n = 64 to 512 and then 29, 28, 26, 27, 25 at every doubling out to 16,384, because κ has reached 99.9% of Szegő's limit and the count has nothing left to grow with.
Structure, and the solver that cannot see itThe 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.