uniqueness-width
At its defaults it draws the widest box a uniqueness proof is available for. Verified half-width against the distance between two roots, both axes logarithmic. The measured widths are 0.05, 0.005, 5·10⁻⁴, 5·10⁻⁵ at separations of 0.1, 0.01, 0.001, 10⁻⁴ — a ratio of 0.5000 at every one of them. The operator verifies exactly up to the point where the box would reach the second root, and a box containing both is refused.
uniqueness-width is one function in lib/figures/krawczyk.js —
verdicts — exactly one root in this box, or none, or nothing can be said. 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.
Verified half-width against the distance between two roots, both axes logarithmic. The measured widths are 0.05, 0.005, 5·10⁻⁴, 5·10⁻⁵ at separations of 0.1, 0.01, 0.001, 10⁻⁴ — a ratio of 0.5000 at every one of them. The operator verifies exactly up to the point where the box would reach the second root, and a box containing both is refused.
deltas: [0.1, 0.01, 0.001, 0.0001]
The arguments are the ones A bound that is proved passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Verified half-width against the distance between two roots, both axes logarithmic. The measured widths are 0.05, 0.005, 5·10⁻⁴, 5·10⁻⁵ at separations of 0.1, 0.01, 0.001, 10⁻⁴ — a ratio of 0.5000 at every one of them. The operator verifies exactly up to the point where the box would reach the second root, and a box containing both is refused.
What it checked while drawing
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.
12 distinct claims across 2 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 a box containing both roots is refused at δ = 0.1 — asserted 4 times
the verified box is narrower than the distance to the second root at δ = 0.1 — asserted 4 times
a separation the arithmetic can resolve
and the verified width is a fixed fraction of the separation
enough separations to show the law
LU is for square matrices
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 66
of 131 generators —
51 print a residual and
15 are exempt with a published reason;
65 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 bound that is proved
Every error statement on this site so far is a measurement of one run. Interval arithmetic makes a different kind of claim — the answer lies in this set, for this input, with no probability attached — and its failure mode is that it returns nothing at all. On a Hilbert system it proves a bound 23 times the error it bounds, and one size later it refuses.
Two errors, and whose fault they areAn answer that is known
Almost every demonstration of numerical error estimates the error by computing the same thing more carefully. The Hilbert matrix does not need that: its inverse is a closed form in integers, so the true answer is available exactly and the error is measured rather than approximated.
The arithmetic underneathProving the answer is in the box
Every other method here computes a number and estimates how wrong it is. This one returns a verdict: there is exactly one solution in this box, or there is none, or — the honest third outcome — nothing can be said. Two of the three are proofs about infinitely many points from finitely many operations.
Eigenvalues, singular values, rankThe plane survives what its vectors do not
At a gap of 10⁻⁹ a perturbation of 10⁻⁶ turns the two eigenvectors through half a radian and turns the plane they span through 7.6·10⁻⁸ — a ratio of six million. Ask for the subspace instead of the vectors and a hopeless computation becomes a well-conditioned one, with no change to the arithmetic.