hierarchy-cost
At its defaults it draws stored entries per row at each level, 400 unknowns and 1192 edges. A rising curve of stored entries per row against level in the hierarchy, beside a flat one for a geometric hierarchy on a grid.
hierarchy-cost is one function in lib/figures/amg.js —
algebraic multigrid — a hierarchy with no grid behind it, and what it costs. 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.
A rising curve of stored entries per row against level in the hierarchy, beside a flat one for a geometric hierarchy on a grid.
extra: 2
The arguments are the ones A hierarchy with no grid behind it 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.
A rising curve of stored entries per row against level in the hierarchy, beside a flat one for a geometric hierarchy on a grid.
extra: 0.5
The arguments are the ones A hierarchy with no grid behind it 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.
A rising curve of stored entries per row against level in the hierarchy, beside a flat one for a geometric hierarchy on a grid.
extra: 3
The arguments are the ones What the symbolic phase can only bound 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.
A rising curve of stored entries per row against level in the hierarchy, beside a flat one for a geometric hierarchy on a grid.
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.
7 distinct claims across 4 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 graph with between half and four extra chords a vertex
and has levels to cost
and the algebraic hierarchy stores more than a geometric one
on a sparse graph it converges quickly
some level is far denser than the matrix it came from
the grid size is one less than a power of two
the hierarchy converges on the problem it is being costed for
Against the rule
It draws a decomposition and prints its residual. It calls
amgSolve,
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 hierarchy with no grid behind it
On a graph Laplacian the algebraic V-cycle converges at 0.199 a cycle, its grid complexity is an unremarkable 3.05, and its operator complexity is 17.7 — one level of forty-one unknowns is entirely dense. The number people quote is the one that does not measure the work.
Iterating, instead of factorisingAn orthogonalisation nobody calls one
Conjugate gradients are derived as a minimisation and behave as an orthogonalisation, which is why the finite-termination property in every textbook is not a property the method has in floating point.
Randomised, and the guarantee that changes kindThe dimension does not appear
A random projection preserves the lengths of a set of vectors to within a distortion that depends on how many vectors there are and not on how many coordinates each one has. That is the fact the whole field rests on, and it is genuinely surprising.
Sparsity, and what elimination costsThe 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 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.