Elimination with and without pivoting, ε = 10⁻¹⁷

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.

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.

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.

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.

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.

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.

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.

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.

5 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.

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 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.