clustered-spectrum
At its defaults it draws the spectrum of c⁻¹t against t's own, n = 64, ρ = 0.8. Two sorted spectra on a linear vertical axis. The unpreconditioned Toeplitz matrix spreads its eigenvalues across its whole range; the preconditioned operator puts 97% of them within a tenth of 1, with 0 below zero.
clustered-spectrum 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 sorted spectra on a linear vertical axis. The unpreconditioned Toeplitz matrix spreads its eigenvalues across its whole range; the preconditioned operator puts 97% of them within a tenth of 1, with 0 below zero.
n: 64, rho: 0.8
The arguments are the ones A limit the matrix never reaches 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 sorted spectra on a linear vertical axis. The unpreconditioned Toeplitz matrix spreads its eigenvalues across its whole range; the preconditioned operator puts 97% of them within a tenth of 1, with 0 below zero.
n: 16, rho: 0.8
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 sorted spectra on a linear vertical axis. The unpreconditioned Toeplitz matrix spreads its eigenvalues across its whole range; the preconditioned operator puts 13% of them within a tenth of 1, with 3 below zero.
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 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 power-of-two size the dense similarity is affordable at
every eigenvalue is accounted for
Jacobi needs a symmetric matrix
matmul shapes agree
the circulant is not singular
the fast transform is given a power-of-two length
the preconditioned spectrum goes negative exactly when the preconditioner does
the similarity transform is symmetric
the Toeplitz corner is one number
the Toeplitz matrix is positive definite
Against the rule
It draws a decomposition and prints its residual. It calls
jacobiEigSym,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
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 limit the matrix never reaches
Szegő's theorem gives a Toeplitz family's condition number in closed form — ((1+ρ)/(1−ρ))², which is 81 at ρ = 0.8. The 8×8 section reaches 52% of it, the 128×128 reaches 98.9%, and none of them ever arrives. A statement about a family is not a statement about the matrix in front of you.
Structure, and the solver that cannot see itA 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 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.