refinement-stalls

At its defaults it draws iterative refinement on a factorisation of a matrix that is not the one being solved. The last member of the sequence, solved with the first member's pivot order. Every other repair in this collection works because the outer loop recomputes the residual from the matrix, and this one does that too — and it does not work. The perturbed pivots are not a rounding, they are a change to the matrix, and the residual that would reveal it is computed with the same badly scaled rows that caused it. Six steps move the backward error from 4.82·10⁻⁹ to 5.46·10⁻¹⁰, a factor of 8.8, and stop. The equilibrated run has nothing to repair: it starts at 6.55·10⁻¹⁷.

refinement-stalls is one function in lib/figures/staticpivot.js — static pivoting — a pivot order kept, and the units that decide whether it may be. 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 last member of the sequence, solved with the first member's pivot order. Every other repair in this collection works because the outer loop recomputes the residual from the matrix, and this one does that too — and it does not work. The perturbed pivots are not a rounding, they are a change to the matrix, and the residual that would reveal it is computed with the same badly scaled rows that caused it. Six steps move the backward error from 4.82·10⁻⁹ to 5.46·10⁻¹⁰, a factor of 8.8, and stop. The equilibrated run has nothing to repair: it starts at 6.55·10⁻¹⁷.

The last member of the sequence, solved with the first member's pivot order. Every other repair in this collection works because the outer loop recomputes the residual from the matrix, and this one does that too — and it does not work. The perturbed pivots are not a rounding, they are a change to the matrix, and the residual that would reveal it is computed with the same badly scaled rows that caused it. Six steps move the backward error from 4.82·10⁻⁹ to 5.46·10⁻¹⁰, a factor of 8.8, and stop. The equilibrated run has nothing to repair: it starts at 6.55·10⁻¹⁷.

The last member of the sequence, solved with the first member's pivot order. Every other repair in this collection works because the outer loop recomputes the residual from the matrix, and this one does that too — and it does not work. The perturbed pivots are not a rounding, they are a change to the matrix, and the residual that would reveal it is computed with the same badly scaled rows that caused it. Six steps move the backward error from 9.55·10⁻¹⁷ to 9.55·10⁻¹⁷, a factor of 1.0, and stop. The equilibrated run has nothing to repair: it starts at 7.21·10⁻¹⁷.

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.

4 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 rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

Across the library: the rule bites on 113 of 238 generators — 98 print a residual and 15 are exempt with a published reason; 125 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.