How far the coefficients can move without changing the fit, degree 9
At its defaults it draws how far the coefficients can move without changing the fit, degree 9. Relative increase in the residual against relative change in the coefficients, along the least determined direction. The residual does not move measurably until the coefficients have changed by more than a factor of one.
flat-valley is one function in lib/figures/lsq.js —
least squares — the projection, the road not to take, and the valley with no bottom. 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.
Relative increase in the residual against relative change in the coefficients, along the least determined direction. The residual does not move measurably until the coefficients have changed by more than a factor of one.
deg: 10
The arguments are the ones A problem with no answer passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Relative increase in the residual against relative change in the coefficients, along the least determined direction. The residual does not move measurably until the coefficients have changed by more than a factor of one.
deg: 9
The arguments are the ones The valley with no bottom passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Relative increase in the residual against relative change in the coefficients, along the least determined direction. The residual does not move measurably until the coefficients have changed by more than a factor of one.
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.
3 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 by orders of magnitude more before it moves in the third
the coefficients can move by a third with the fit unchanged in the sixth digit
which is what a condition number of this size means
Against the rule
It calls a factoriser without drawing a factorisation
(svd, lstsqQR),
so the rule is written down as not applying, with the reason:
uses the SVD for the worst-conditioned direction only
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.
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.
A problem with no answer
If two matrices share a null vector then det(A − λB) is identically zero and every λ is an eigenvalue, which means none of them is. Perturb such a pencil by a ten-billionth and a solver returns six numbers with residuals below 10⁻⁹. Change the seed and it returns six different numbers, spread over forty-four, with residuals just as small.
Least squares, and the road not to takeThe valley with no bottom
A degree-nine fit's coefficients can be moved by a third of their own size before the residual changes in the sixth significant figure. The arithmetic did not lose those digits. The data never contained them.