strang-sign-change
At its defaults it draws conjugate gradient steps either side of the preconditioner's sign change, ρ = 0.9. Two step counts against the size, on logarithmic axes. Without a preconditioner the count climbs steadily. With Strang's circulant preconditioner it is worse than useless while the preconditioner is indefinite — its smallest eigenvalue is -0.4005 at n = 16 — and collapses once the sign changes, between n = 32 and n = 64.
strang-sign-change is one function in lib/figures/structure.js —
structure — the matrix that is one row, and the solver that cannot see it. 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.
Two step counts against the size, on logarithmic axes. Without a preconditioner the count climbs steadily. With Strang's circulant preconditioner it is worse than useless while the preconditioner is indefinite — its smallest eigenvalue is -0.4005 at n = 16 — and collapses once the sign changes, between n = 32 and n = 64.
rho: 0.9
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.
Two step counts against the size, on logarithmic axes. Without a preconditioner the count climbs steadily. With Strang's circulant preconditioner it is worse than useless while the preconditioner is indefinite — its smallest eigenvalue is -0.4005 at n = 16 — and collapses once the sign changes, between n = 32 and n = 64.
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 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 smallest eigenvalue of C does not fall at n = 16 — asserted 5 times
the unpreconditioned count does not fall at n = 32 — asserted 4 times
where C is definite the preconditioned count does not climb at n = 128 — asserted 2 times
a correlation where the sign change is inside the range drawn
and climbs over the range as a whole
and where it is not, the repair is not an improvement
the circulant is not singular
the fast transform is given a power-of-two length
the preconditioner is definite somewhere in the range drawn
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 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 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.