The diagonal of a pivoted R against the singular values, on a 40×40 Kahan matrix at c = 0.5

At its defaults it draws the diagonal of a pivoted r against the singular values, on a 40×40 kahan matrix at c = 0.5. Two curves down the index. The diagonal of R decays at one constant rate to 0.00366; the singular values follow it and then fall away to 9.07·10⁻¹⁰ at the last one. Column pivoting made no interchange at all, because every trailing column norm is equal at every step — the rule had nothing to choose between.

rank-diagonal is one function in lib/figures/rrqr.js — the cheap rank — a greedy pivot rule, and the matrix it has nothing to choose on. 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.

Two curves down the index. The diagonal of R decays at one constant rate to 0.00366; the singular values follow it and then fall away to 9.07·10⁻¹⁰ at the last one. Column pivoting made no interchange at all, because every trailing column norm is equal at every step — the rule had nothing to choose between.

Two curves down the index, lying on top of each other. Both fall by more than five orders of magnitude between index 12 and index 13, so both report rank 12 at any threshold placed in the gap.

Two curves down the index, lying on top of each other. Both fall by more than five orders of magnitude between index 4 and index 5, so both report rank 4 at any threshold placed in the gap.

Two curves down the index, lying on top of each other. Both fall by more than five orders of magnitude between index 24 and index 25, so both report rank 24 at any threshold placed in the gap.

Two curves down the index. The diagonal of R decays at one constant rate to 0.00366; the singular values follow it and then fall away to 9.07·10⁻¹⁰ at the last one. Column pivoting made no interchange at all, because every trailing column norm is equal at every step — the rule had nothing to choose between.

Two curves down the index. The diagonal of R decays at one constant rate to 0.451; the singular values follow it and then fall away to 7.05·10⁻⁴ at the last one. Column pivoting made no interchange at all, because every trailing column norm is equal at every step — the rule had nothing to choose between.

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.

75 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 draws a decomposition and prints its residual. It calls svd, qrColumnPivoted, 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.