perturbed-eigenvalues
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: -8
The arguments are the ones A spectral radius that grows first 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.
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: -6
The arguments are the ones An eigenvalue one vector cannot see 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.
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 10⁻⁶. The normal matrix's eigenvalues stay within 8.86·10⁻⁷ of where they were — the size of the perturbation — and the bidiagonal's spread out to 0.151, a factor of 1.7·10⁵ further on the same data.
logEps: -10
The arguments are the ones An eigenvalue that arrives twice 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.
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 10⁻¹⁰. The normal matrix's eigenvalues stay within 7.43·10⁻¹¹ of where they were — the size of the perturbation — and the bidiagonal's spread out to 0.0325, a factor of 4.4·10⁸ further on the same data.
logEps: -4
The arguments are the ones The eigenvalues that are not there 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.
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 drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run 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 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.
9 distinct claims across 6 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 σ_min(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,
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 90
of 174 generators —
75 print a residual and
15 are exempt with a published reason;
84 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 spectral radius that grows first
ρ(A) below one guarantees that the powers of A go to zero and says nothing about what they do on the way. Here they rise by a factor of twenty thousand before turning over, and the peak is bracketed above and below by a constant computed from the resolvent norms outside the unit circle — two routes to one number, one through the plane and one through the powers.
Eigenvalues, singular values, rankAn eigenvalue one vector cannot see
A matrix with an exactly doubled eigenvalue at 10. Twelve Lanczos steps find it once; twenty-four find it once, on a Krylov space of dimension 23 in a 24-dimensional problem. A block of two vectors finds it twice. This is not slow convergence — the second copy is not in the space.
Eigenvalues, singular values, rankAn eigenvalue that arrives twice
A matrix with forty distinct eigenvalues, handed to Lanczos for eighty steps, returns twenty-five extra copies of thirteen of them — the largest arriving five times. Every copy is accurate to 1.9·10⁻⁸ relative. No arithmetic error was made, nothing overflowed, and a caller counting eigenvalues gets the wrong multiplicity from a computation in which no individual number is wrong.
Iterating, instead of factorisingThe direction the diffusion does not go
Streamline diffusion adds τbbᵀ, a rank-one tensor that annihilates every direction across the flow. That is the design. The price is 18, 50 and 48 nodes where the computed solution leaves the interval the equation guarantees — and half a coefficient of crosswind diffusion halves the error at every angle while costing exactly nothing where the scheme was exact.
Eigenvalues, singular values, rankThe eigenvalues that are not there
For a normal matrix the resolvent norm is exactly one over the distance to the nearest eigenvalue, so a picture of it carries nothing the spectrum did not. Move one entry above the diagonal and the region a perturbation of 10⁻⁸ can put an eigenvalue into stops being a disc and reaches out past the unit circle, while every eigenvalue stays at 0.8.
Eigenvalues, singular values, rankThe gap decides the eigenvector
A symmetric matrix's eigenvalues move by at most the size of the perturbation, whatever the spectrum looks like. Its eigenvectors are governed by a completely different quantity — the distance to the neighbouring eigenvalue — and at a gap of 10⁻⁹ the same perturbation turns them through 27°.