three-counters
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 drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run 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: 144
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.
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: 400
The arguments are the ones The same arithmetic at a different price 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 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 3.41 apart. The factorisations they produce are identical to the last bit.
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.
29 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.
and less traffic at n = 24 — asserted 6 times
and the same factorisation at n = 24 — asserted 6 times
and the same pivots at n = 24 — asserted 6 times
the same operations at n = 24 — asserted 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 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 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.
Elimination, and the swapElimination is a sequence of choices
Gaussian elimination is taught as a procedure with no decisions in it. There is one decision at every step — which row to use — and every stability property the algorithm has comes from making it well.
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.