Where 24 perturbations of size 10⁻⁸ put the eigenvalues of two 6×6 matrices with the same spectrum
At its defaults it draws where 24 perturbations of size 10⁻⁸ put the eigenvalues of two 6×6 matrices with the same spectrum. Two clouds of eigenvalues in the complex plane. Both matrices have every eigenvalue at 0.8 exactly; both were perturbed by the same 24 random matrices of norm 10⁻⁸. The normal matrix's eigenvalues stay within 8.86·10⁻⁹ of where they were — the size of the perturbation — and the bidiagonal's spread out to 0.0701, a factor of 7.9·10⁶ further on the same data.
perturbed-eigenvalues is one function in lib/figures/pseudo.js —
pseudospectra — where the eigenvalues would be, and what the powers do first. 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.
Two clouds of eigenvalues in the complex plane. Both matrices have every eigenvalue at 0.8 exactly; both were perturbed by the same 24 random matrices of norm 10⁻⁸. The normal matrix's eigenvalues stay within 8.86·10⁻⁹ of where they were — the size of the perturbation — and the bidiagonal's spread out to 0.0701, a factor of 7.9·10⁶ further on the same data.
logEps: -4
The arguments are the ones The eigenvalues that are not there passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two clouds of eigenvalues in the complex plane. Both matrices have every eigenvalue at 0.8 exactly; both were perturbed by the same 24 random matrices of norm 10⁻⁴. The normal matrix's eigenvalues stay within 8.86·10⁻⁵ of where they were — the size of the perturbation — and the bidiagonal's spread out to 0.329, a factor of 3710 further on the same data.
logEps: -12
The arguments are the ones The eigenvalues that are not there passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two clouds of eigenvalues in the complex plane. Both matrices have every eigenvalue at 0.8 exactly; both were perturbed by the same 24 random matrices of norm 10⁻¹². The normal matrix's eigenvalues stay within 6.51·10⁻¹³ of where they were — the size of the perturbation — and the bidiagonal's spread out to 0.0151, a factor of 2.3·10¹⁰ further on the same data.
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.
9 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 perturbation at or below the level of a data error
a power of ten rather than an exponent literal
a size the repeated Schur forms can afford
and every one of them satisfies σₘᵢₙ(λ̃I − A) ≤ ε, which is the definition
and the bidiagonal's move orders of magnitude further
enough perturbations to see a cloud
matmul shapes agree
on the same perturbations
the normal matrix's eigenvalues move by the size of the perturbation
Against the rule
It draws a decomposition and prints its residual. It calls
spectrum, sigmaMinAt, spectrum,
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.
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.