The same arithmetic, twice, with M = 144 words of fast memory
At its defaults it draws the same arithmetic, twice, with m = 144 words of fast memory. Three counters against the matrix size on logarithmic axes. The operation count is one curve, because both orderings perform exactly the same operations — 338,120 of them at n = 80. The words moved between fast and slow memory are two curves a factor of 3.24 apart. The factorisations they produce are identical to the last bit.
three-counters is one function in lib/figures/cost.js —
cost — where the flop count stopped predicting the time. 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 counters against the matrix size on logarithmic axes. The operation count is one curve, because both orderings perform exactly the same operations — 338,120 of them at n = 80. The words moved between fast and slow memory are two curves a factor of 3.24 apart. The factorisations they produce are identical to the last bit.
fast: 100
The arguments are the ones A block size is a property of the machine passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three counters against the matrix size on logarithmic axes. The operation count is one curve, because both orderings perform exactly the same operations — 338,120 of them at n = 80. The words moved between fast and slow memory are two curves a factor of 3.89 apart. The factorisations they produce are identical to the last bit.
fast: 256
The arguments are the ones A block size is a property of the machine passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three counters against the matrix size on logarithmic axes. The operation count is one curve, because both orderings perform exactly the same operations — 338,120 of them at n = 80. The words moved between fast and slow memory are two curves a factor of 2.91 apart. The factorisations they produce are identical to the last bit.
fast: 196
The arguments are the ones Memory bought with messages passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three counters against the matrix size on logarithmic axes. The operation count is one curve, because both orderings perform exactly the same operations — 338,120 of them at n = 80. The words moved between fast and slow memory are two curves a factor of 2.67 apart. The factorisations they produce are identical to the last bit.
fast: 144
The arguments are the ones The same arithmetic at a different price passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three counters against the matrix size on logarithmic axes. The operation count is one curve, because both orderings perform exactly the same operations — 338,120 of them at n = 80. The words moved between fast and slow memory are two curves a factor of 3.24 apart. The factorisations they produce are identical to the last bit.
fast: 36
The arguments are the ones The same arithmetic at a different price passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three counters against the matrix size on logarithmic axes. The operation count is one curve, because both orderings perform exactly the same operations — 338,120 of them at n = 80. The words moved between fast and slow memory are two curves a factor of 2.74 apart. The factorisations they produce are identical to the last bit.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks 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.
29 distinct claims across 6 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 less traffic at n = 24 — checked 6 times
and the same factorisation at n = 24 — checked 6 times
and the same pivots at n = 24 — checked 6 times
the same operations at n = 24 — checked 6 times
a fast memory small enough for the matrix not to fit in it
at every size drawn
matmul shapes agree
the block fits the matrix
there are sizes the matrix does not fit in fast memory at
Against the rule
It draws a decomposition and prints its residual. It calls
blockedLU,
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 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 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 timeMemory bought with messages
Holding four copies of the data instead of one is supposed to cut a matrix multiplication's communication by √4. Measured on a machine of 64 processors it costs 14% more traffic; at 576 it saves 44%, which is 72% of what the law promises. The memory is exactly four times, and that part is not asymptotic.
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.