rounds-vs-words
At its defaults it draws communication for a 512×12 factorisation on 16 processors. Two counts for the same factorisation. Rounds on the critical path: 48 for the column sweep, 4 for the reduction tree, 4 for Cholesky QR — a factor of n between the first and the other two. Words sent: 1170, 1170 and 2160 — the sweep and the tree send the same number, and the method with the fewest rounds sends the most.
rounds-vs-words is one function in lib/figures/parallel.js —
messages — the count words cannot make, and what the fewest of them 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.
Two counts for the same factorisation. Rounds on the critical path: 48 for the column sweep, 4 for the reduction tree, 4 for Cholesky QR — a factor of n between the first and the other two. Words sent: 1170, 1170 and 2160 — the sweep and the tree send the same number, and the method with the fewest rounds sends the most.
p: 32
The arguments are the ones A block size is a property of the machine 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 counts for the same factorisation. Rounds on the critical path: 60 for the column sweep, 5 for the reduction tree, 5 for Cholesky QR — a factor of n between the first and the other two. Words sent: 2418, 2418 and 4464 — the sweep and the tree send the same number, and the method with the fewest rounds sends the most.
p: 16
The arguments are the ones A reduction that changes the order 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 counts for the same factorisation. Rounds on the critical path: 48 for the column sweep, 4 for the reduction tree, 4 for Cholesky QR — a factor of n between the first and the other two. Words sent: 1170, 1170 and 2160 — the sweep and the tree send the same number, and the method with the fewest rounds sends the most.
p: 8
The arguments are the ones An eigenvalue one vector cannot see 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 counts for the same factorisation. Rounds on the critical path: 36 for the column sweep, 3 for the reduction tree, 3 for Cholesky QR — a factor of n between the first and the other two. Words sent: 546, 546 and 1008 — the sweep and the tree send the same number, and the method with the fewest rounds sends the most.
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.
8 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 processor count the reduction tree is defined for
and Cholesky QR takes one reduction
each row block is at least as tall as it is wide
each slice is at least as tall as it is wide
matmul shapes agree
the sweep takes n log₂p rounds
the tree takes log₂p
while the one-reduction method sends more words than the tree
Against the rule
It draws a decomposition and prints its residual. It calls
choleskyQR, tsqrDistributed, householderDistributed,
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 block size is a property of the machine
Three lines of counting say the best block size is √(M/3). Scanned over every integer at five fast memories, the measured optimum is √M − 2 — exactly, at all five. The count has the right scaling and the wrong constant, low by a factor of 1.56, and the wrong form: the answer is affine in √M rather than proportional to it.
Where the flop count stopped predicting the timeA reduction that changes the order
A tall-skinny QR computed as a tree of independent block factorisations touches a 512×12 matrix once instead of twelve times, computes a completely different sequence of roundings from the sweep it replaces, and returns ‖AᵀA − RᵀR‖/‖AᵀA‖ = 1.65·10⁻¹⁵ against the sweep's 9.95·10⁻¹⁵. On the same matrix classical Gram–Schmidt returns 4.6·10⁻¹⁰.
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.
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.
Where the flop count stopped predicting the timeThe same arithmetic at a different price
A blocked and an unblocked elimination perform 72,568 operations each — the same operations, associated differently — choose the same pivots, and return a factorisation identical to the last bit: ‖PA − LU‖/‖A‖ = 4.487946226420872·10⁻¹⁶ in both. One of them moves 41,332 words between fast and slow memory and the other moves 19,476.