chan-vs-strang
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: 0.95
The arguments are the ones A preconditioner that changes sign 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 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: 0.9
The arguments are the ones Changing the condition number on purpose 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 matrix size, drawn on a logarithmic size axis. Unpreconditioned: 20, 35, 54, 82, 117. With the wrapped circulant: 20, 45, 26, 7, 4. With the averaged one: 7, 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 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.
17 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.
and beats no preconditioner at n = 16 — asserted 5 times
the averaged circulant is definite at n = 16 — asserted 5 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 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 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 factorisingChanging the condition number on purpose
Preconditioning is usually introduced as a trick that makes an iteration converge faster. It is not a trick. It is solving a different system with the same solution and a condition number chosen rather than inherited, and the new condition number is computable.
Sparsity, and what elimination costsStructure and stability stop being separable
The sparsest variable to eliminate on this matrix has a diagonal entry of 10⁻¹². Eliminating it produces the smaller factor, reproduces the matrix to 3.8·10⁻¹⁷ — better than pivoting does — and returns an answer wrong in the fifth digit.
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.
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%.