arrowhead-two-ways
At its defaults it draws the arrowhead matrix, eliminated from each end. Three sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.
arrowhead-two-ways is one function in lib/figures/sparse.js —
sparsity — fill counted two ways, and the ordering that decides the memory. 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.
Three sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.
n: 22
The arguments are the ones The factor is not sparse 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.
Three sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.
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.
4 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.
and both factorisations are exact
matmul shapes agree
tip first fills the factor completely
tip last creates no fill at all
Against the rule
It draws a decomposition and prints its residual. It calls
cholesky,
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.
The factor is not sparse
A sparse matrix has a factor that is not sparse, and the gap between them is the entire reason iterative methods exist. The entries elimination creates can be counted before any arithmetic runs, from the graph alone.
Sparsity, and what elimination costsThe order decides the memory
Four elimination orderings on one matrix give factors of 1,739, 1,354, 1,413 and 1,026 entries. All four factorisations are exact, all four return the same answer, and the one with the better asymptotics is not the one that wins.
Sparsity, and what elimination costsTwo ends of the same arrow
One matrix, one row moved from the front of the elimination order to the back, and the factor goes from completely dense to no fill at all. Both factorisations are exact to rounding, and nothing numerical chose between them.
Sparsity, and what elimination costsWhat the symbolic phase can only bound
Without pivoting, the fill can be computed from the graph and the count is exact — 233 predicted, 233 measured. With pivoting it is 233 predicted and 242 measured, and what survives is a bound that is right at every threshold and loose by 1.7 times at the largest grid drawn.