pivot-failure
At its defaults it draws elimination with and without pivoting, ε = 10⁻¹⁷. The same two-by-two system solved twice. With a row swap the answer is exact; without one the upper triangular factor contains an entry of order one over epsilon and the second component of the answer is wrong.
pivot-failure is one function in lib/figures/elim.js —
elimination — the swap, the growth factor, and the matrix with a known answer. 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.
The same two-by-two system solved twice. With a row swap the answer is exact; without one the upper triangular factor contains an entry of order one over epsilon and the second component of the answer is wrong.
logEps: -12
The arguments are the ones Structure and stability stop being separable 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.
The same two-by-two system solved twice. With a row swap the answer is exact; without one the upper triangular factor contains an entry of order one over epsilon and the second component of the answer is wrong.
logEps: -17
The arguments are the ones The factor is not sparse 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.
The same two-by-two system solved twice. With a row swap the answer is exact; without one the upper triangular factor contains an entry of order one over epsilon and the second component of the answer is wrong.
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.
5 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.
and the unpivoted one is worse
elimination without the swap multiplies the entries by 1/ε agree
LU is for square matrices
the closed-form solution, written two ways agree
the pivoted solve is backward stable
Against the rule
It calls a factoriser without drawing a factorisation
(luFactor, solve),
so the rule is written down as not applying, with the reason:
prints backward and forward error for both runs, which is the stronger statement here
The exemption list is the interesting half of the rule rather than an escape hatch — it is
where a decision about a figure had to be argued in one line. residualcheck
refuses an exemption that is not doing work, and rejected ten of the fifteen written for the
expansion's figures on exactly that ground: a figure whose vertical axis is a residual
satisfies the rule by construction, and touching a factoriser does not by itself require an
entry.
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.
Structure and stability stop being separable
The sparsest variable to eliminate on this matrix has a diagonal entry of 10⁻¹². Eliminating it produces the smaller factor, reproduces the matrix to 3.8·10⁻¹⁷ — better than pivoting does — and returns an answer wrong in the fifth digit.
Sparsity, and what elimination costsThe factor is not sparse
A sparse matrix has a factor that is not sparse, and the gap between them is the entire reason iterative methods exist. The entries elimination creates can be counted before any arithmetic runs, from the graph alone.
Elimination, and the swapThe swap that is not optional
Run elimination without a row interchange on a matrix that needs one and nothing announces a failure. There is no division by zero, no warning, and an answer of the right shape. It is simply wrong, and how wrong depends on a number you did not look at.
Sparsity, and what elimination costsTwo ends of the same arrow
One matrix, one row moved from the front of the elimination order to the back, and the factor goes from completely dense to no fill at all. Both factorisations are exact to rounding, and nothing numerical chose between them.