Iterative refinement on a factorisation of a matrix that is not the one being solved
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.
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 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.
decades: 8
The arguments are the ones The gap refinement can close passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
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 8.23·10⁻¹⁰ to 2.43·10⁻¹⁴, a factor of 33820.6, and stop. The equilibrated run has nothing to repair: it starts at 5.74·10⁻¹⁷.
decades: 9
The arguments are the ones The order that was right last time passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
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.
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.
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.
a change of units inside the range the family is defined over
and moves it by less than a factor of thirty in six steps
both runs have a history to draw
refinement does not bring the damaged run to the working precision
Against the rule
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 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.
The gap refinement can close
Multiplying by a computed inverse is not backward stable, and refinement at the working precision repairs it. That much is settled. The claim beside it — that the forward error does not move — was read at one conditioning and four corrections too late. Swept over ten, it moves at every one, and it lands on the LU route's own number after a single correction.
Sparsity, and what elimination costsThe order that was right last time
A pivot order computed once and reused across a sequence saves the symbolic phase, and the price is that a pivot which was large may now be small. Replacing it with √u·‖A‖ costs eight orders of backward error and iterative refinement recovers a factor of 8.8 of them. Divide each row by its largest entry first and the same reuse costs nothing at all.