gap-cost
At its defaults it draws separating two eigenvalues, against how close they are. Iterations against the gap between the two largest eigenvalues, on a logarithmic gap axis. The single-vector method needs 17, 18, 20, 22 steps as the gap closes through four decades. The block of two needs 11 at every gap — and spends 22 products with A doing it, which is no less arithmetic. What it saves is synchronisations.
gap-cost is one function in lib/figures/restart.js —
restarting and blocks — a filter on the starting vector, and what one vector cannot see. 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.
Iterations against the gap between the two largest eigenvalues, on a logarithmic gap axis. The single-vector method needs 17, 18, 20, 22 steps as the gap closes through four decades. The block of two needs 11 at every gap — and spends 22 products with A doing it, which is no less arithmetic. What it saves is synchronisations.
n: 24
The arguments are the ones A condition number for one eigenvalue 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.
Iterations against the gap between the two largest eigenvalues, on a logarithmic gap axis. The single-vector method needs 17, 18, 20, 22 steps as the gap closes through four decades. The block of two needs 11 at every gap — and spends 22 products with A doing it, which is no less arithmetic. What it saves is synchronisations.
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 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.
both methods separate a gap of 0.1 — asserted 4 times
a size the search over step counts is affordable at
and the block's does not move
enough gaps to show a trend
Jacobi needs a symmetric matrix
matmul shapes agree
the single-vector iteration count grows as the gap closes
Against the rule
It draws a decomposition and prints its residual. It calls
blockLanczos,
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 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 condition number for one eigenvalue
In the symmetric case every eigenvalue has condition number exactly one. In this four-by-four matrix two of them have condition number 100.005 and the other two have exactly 1, and the number belongs to the eigenvalue rather than to the matrix.
Eigenvalues, singular values, rankAn eigenvalue one vector cannot see
A matrix with an exactly doubled eigenvalue at 10. Twelve Lanczos steps find it once; twenty-four find it once, on a Krylov space of dimension 23 in a 24-dimensional problem. A block of two vectors finds it twice. This is not slow convergence — the second copy is not in the space.
Eigenvalues, singular values, rankThe gap decides the eigenvector
A symmetric matrix's eigenvalues move by at most the size of the perturbation, whatever the spectrum looks like. Its eigenvectors are governed by a completely different quantity — the distance to the neighbouring eigenvalue — and at a gap of 10⁻⁹ the same perturbation turns them through 27°.
Where the flop count stopped predicting the timeThe message and the word
Three factorisations of one matrix on sixteen processors: 48 communication rounds, 4, and 4. The words sent are 1,170, 1,170 and 2,160 — so the method with the fewest rounds sends the most words, and the count that separates the three is the one no operation count can see.