precision-asymmetry
At its defaults it draws preconditioned conjugate gradients with one part of it rounded. Iteration count against significand bits, on the 100-unknown model problem. Rounding the preconditioner's output takes the count from 18 to 35 and leaves the answer correct to 8.8·10⁻¹³ throughout. Rounding the working arithmetic instead leaves the count at 18–300 and takes the answer to 0.159. The horizontal line is the 39 steps the unpreconditioned method takes.
precision-asymmetry is one function in lib/figures/mixedcg.js —
precision inside an iteration — what may be rounded, and what may not. 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.
Iteration count against significand bits, on the 100-unknown model problem. Rounding the preconditioner's output takes the count from 18 to 35 and leaves the answer correct to 8.8·10⁻¹³ throughout. Rounding the working arithmetic instead leaves the count at 18–300 and takes the answer to 0.159. The horizontal line is the 39 steps the unpreconditioned method takes.
lowest: 6
The arguments are the ones A norm that overflows before it is a norm 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.
Iteration count against significand bits, on the 100-unknown model problem. Rounding the preconditioner's output takes the count from 18 to 22 and leaves the answer correct to 8.9·10⁻¹³ throughout. Rounding the working arithmetic instead leaves the count at 18–40 and takes the answer to 0.0179. The horizontal line is the 39 steps the unpreconditioned method takes.
lowest: 3
The arguments are the ones Buying the accuracy back 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.
Iteration count against significand bits, on the 100-unknown model problem. Rounding the preconditioner's output takes the count from 18 to 35 and leaves the answer correct to 8.8·10⁻¹³ throughout. Rounding the working arithmetic instead leaves the count at 18–300 and takes the answer to 0.159. The horizontal line is the 39 steps the unpreconditioned method takes.
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.
16 distinct claims across 3 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.
the answer survives a 53-bit preconditioner — asserted 9 times
a grid the dense reference solve is affordable on
a precision the figure has something to show below
and the step count is what rounding the preconditioner costs
LU is for square matrices
matmul shapes agree
the incomplete Cholesky factorisation exists on the model problem
while rounding the working arithmetic costs the answer
Against the rule
It draws a decomposition and prints its residual. It calls
withRoundedPreconditioner, withRoundedWorkingPrecision,
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 norm that overflows before it is a norm
The vector of sixteen thousands has a Euclidean norm of 4,000, which fp16 represents exactly. Written as the square root of the sum of squares it returns infinity, because squaring doubles the exponent — and the expression costs half the format's range on the one computation every iterative method performs at every step.
The arithmetic underneathBuying the accuracy back
Factorise in single precision, then correct the answer using residuals computed in double, and the result is what a full double-precision solve would have given. Compute those residuals in single instead and the identical algorithm, at identical cost, recovers nothing.
Iterating, instead of factorisingChanging the condition number on purpose
Preconditioning is usually introduced as a trick that makes an iteration converge faster. It is not a trick. It is solving a different system with the same solution and a condition number chosen rather than inherited, and the new condition number is computable.
Methods that were designed apartThe part of a solver that may be rounded
A preconditioner computed and applied with a three-bit significand still returns thirteen correct digits — it costs seventeen extra iterations and nothing else. Round the working arithmetic instead and the step count barely moves while the answer loses exactly the digits the format dropped.
The arithmetic underneathWhere the hardware went
bfloat16 carries eight mantissa bits, which puts its refinement threshold at a condition number of 256. That is not an exotic matrix. It is an ordinary one, and past it the method still improves the answer by a factor of four hundred while getting nowhere near a usable one.