Conjugate gradient steps on the ρ = 0.95 Toeplitz family
At its defaults it draws conjugate gradient steps on the ρ = 0.95 toeplitz family. Iteration count against the matrix size, drawn on a logarithmic size axis. Unpreconditioned: 20, 37, 59, 99, 149. With the wrapped circulant: 21, 55, 117, 122, 9. With the averaged one: 7, 8, 9, 10, 10, which is the O(1) the theory promises and the wrapped one does not deliver until its smallest eigenvalue has become positive.
chan-vs-strang 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.
Iteration count against the matrix size, drawn on a logarithmic size axis. Unpreconditioned: 20, 37, 59, 99, 149. With the wrapped circulant: 21, 55, 117, 122, 9. With the averaged one: 7, 8, 9, 10, 10, which is the O(1) the theory promises and the wrapped one does not deliver until its smallest eigenvalue has become positive.
rho: 0.6
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.
Iteration count against the matrix size, drawn on a logarithmic size axis. Unpreconditioned: 16, 25, 33, 37, 40. With the wrapped circulant: 9, 5, 4, 3, 3. With the averaged one: 8, 8, 7, 7, 6, which is the O(1) the theory promises and the wrapped one does not deliver until its smallest eigenvalue has become positive.
rho: 0.5, sizes: [16, 32, 64, 128, 256, 512]
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.
Iteration count against the matrix size, drawn on a logarithmic size axis. Unpreconditioned: 15, 22, 29, 28, 30, 30. With the wrapped circulant: 7, 4, 3, 3, 3, 3. With the averaged one: 8, 8, 7, 6, 6, 5, which is the O(1) the theory promises and the wrapped one does not deliver until its smallest eigenvalue has become positive.
rho: 0.7, sizes: [16, 32, 64, 128, 256, 512]
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.
Iteration count against the matrix size, drawn on a logarithmic size axis. Unpreconditioned: 19, 29, 38, 46, 56, 57. With the wrapped circulant: 13, 7, 4, 3, 3, 3. With the averaged one: 8, 9, 8, 7, 7, 6, which is the O(1) the theory promises and the wrapped one does not deliver until its smallest eigenvalue has become positive.
rho: 0.8, sizes: [16, 32, 64, 128, 256, 512]
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.
Iteration count against the matrix size, drawn on a logarithmic size axis. Unpreconditioned: 18, 30, 45, 62, 77, 86. With the wrapped circulant: 18, 14, 6, 4, 3, 3. With the averaged one: 8, 9, 8, 8, 7, 6, which is the O(1) the theory promises and the wrapped one does not deliver until its smallest eigenvalue has become positive.
rho: 0.95, sizes: [16, 32, 64, 128, 256, 512]
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.
Iteration count against the matrix size, drawn on a logarithmic size axis. Unpreconditioned: 20, 37, 59, 99, 149, 219. With the wrapped circulant: 21, 55, 117, 122, 9, 5. With the averaged one: 7, 8, 9, 10, 10, 9, which is the O(1) the theory promises and the wrapped one does not deliver until its smallest eigenvalue has become positive.
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.
19 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.
and beats no preconditioner at n = 16 — checked 6 times
the averaged circulant is definite at n = 16 — checked 6 times
a correlation the family is conditioned at
enough sizes to show a trend
the averaged circulant's step count barely moves
the circulant is not singular
the fast transform is given a power-of-two length
the Toeplitz corner is one number
while the unpreconditioned count grows with the size
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.