How often Cholesky still calls a 12×12 matrix positive definite, against λₘᵢₙ/λₘₐₓ in units of the format's own roundoff
At its defaults it draws how often cholesky still calls a 12×12 matrix positive definite, against λₘᵢₙ/λₘₐₓ in units of the format's own roundoff. Three curves, one per precision, of the share of 24 seeded matrices on which the factorisation succeeds. Measured in units of each format's unit roundoff the three lie almost on top of one another, with the edge — the smallest ratio at which every seed succeeds — at 1.0, 1.0, 1.8 times u. The absolute thresholds are 6·10⁻⁸, 2.3·10⁻¹⁰ and 2·10⁻¹⁶: nine orders of magnitude apart, and the same number in the format's own units.
definiteness-edge is one function in lib/figures/symmetric.js —
symmetric eliminations — a growth factor of exactly one, and a diagonal that is zero. 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 curves, one per precision, of the share of 24 seeded matrices on which the factorisation succeeds. Measured in units of each format's unit roundoff the three lie almost on top of one another, with the edge — the smallest ratio at which every seed succeeds — at 1.0, 1.0, 1.8 times u. The absolute thresholds are 6·10⁻⁸, 2.3·10⁻¹⁰ and 2·10⁻¹⁶: nine orders of magnitude apart, and the same number in the format's own units.
n: 12
The arguments are the ones A factorisation with nothing to pivot for passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three curves, one per precision, of the share of 24 seeded matrices on which the factorisation succeeds. Measured in units of each format's unit roundoff the three lie almost on top of one another, with the edge — the smallest ratio at which every seed succeeds — at 1.0, 1.0, 1.8 times u. The absolute thresholds are 6·10⁻⁸, 2.3·10⁻¹⁰ and 2·10⁻¹⁶: nine orders of magnitude apart, and the same number in the format's own units.
n: 20
The arguments are the ones A factorisation with nothing to pivot for passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three curves, one per precision, of the share of 24 seeded matrices on which the factorisation succeeds. Measured in units of each format's unit roundoff the three lie almost on top of one another, with the edge — the smallest ratio at which every seed succeeds — at 1.0, 1.0, 1.0 times u. The absolute thresholds are 6·10⁻⁸, 2.3·10⁻¹⁰ and 1.1·10⁻¹⁶: nine orders of magnitude apart, and the same number in the format's own units.
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.
10 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 edge at 24 bits is a small multiple of that format's u — checked 3 times
a size the seeded sweep can afford
and does not succeed everywhere at the bottom of it
and far below it the answer is a coin flip
matmul shapes agree
the test succeeds everywhere at the top of the grid
while the absolute thresholds span the precisions
with the same constant in front of u at every precision
Against the rule
It draws a decomposition and prints its residual. It calls
definitenessEdge,
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.