Generator

eigen-sensitivity

One function in the spectra library, called 14 times across 13 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 36 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws how far a perturbation of size ε moves an eigenvalue, 8×8. A log–log plot of eigenvalue movement against perturbation size. The symmetric case lies on a line of slope one; the non-symmetric case lies on a line of slope one eighth, and at a perturbation of ten to the minus sixteen it has already moved by a hundredth.

eigen-sensitivity is one function in lib/figures/spectra.js — spectra — sensitivity, the symmetric easy case, and rank as a decision. 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.

A log–log plot of eigenvalue movement against perturbation size. The symmetric case lies on a line of slope one; the non-symmetric case lies on a line of slope one eighth, and at a perturbation of ten to the minus sixteen it has already moved by a hundredth.

`n: 8`

The arguments are the ones A condition number for one eigenvalue 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.

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.

36 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.

Weyl's bound holds at ε = 0.001 — asserted 5 times

a rounding-level perturbation moves the non-symmetric spectrum by more than a thousandth

and it is worse than the symmetric case everywhere on this range

Jacobi needs a symmetric matrix

matmul shapes agree

the 8-th root of ε is an eigenvalue of the perturbed Jordan block

Weyl's bound holds at ε = 10⁻¹⁰

Weyl's bound holds at ε = 10⁻¹¹

Weyl's bound holds at ε = 10⁻¹²

Weyl's bound holds at ε = 10⁻¹³

Weyl's bound holds at ε = 10⁻¹⁴

Weyl's bound holds at ε = 10⁻¹⁵

Weyl's bound holds at ε = 10⁻¹⁶

Weyl's bound holds at ε = 10⁻⁴

Weyl's bound holds at ε = 10⁻⁵

Weyl's bound holds at ε = 10⁻⁶

Weyl's bound holds at ε = 10⁻⁷

Weyl's bound holds at ε = 10⁻⁸

Weyl's bound holds at ε = 10⁻⁹

Weyl's bound holds at ε = 3.2·10⁻¹⁰

Weyl's bound holds at ε = 3.2·10⁻¹¹

Weyl's bound holds at ε = 3.2·10⁻¹²

Weyl's bound holds at ε = 3.2·10⁻¹³

Weyl's bound holds at ε = 3.2·10⁻¹⁴

Weyl's bound holds at ε = 3.2·10⁻¹⁵

Weyl's bound holds at ε = 3.2·10⁻¹⁶

Weyl's bound holds at ε = 3.2·10⁻⁴

Weyl's bound holds at ε = 3.2·10⁻⁵

Weyl's bound holds at ε = 3.2·10⁻⁶

Weyl's bound holds at ε = 3.2·10⁻⁷

Weyl's bound holds at ε = 3.2·10⁻⁸

Weyl's bound holds at ε = 3.2·10⁻⁹

Against the rule

It calls a factoriser without drawing a factorisation (jacobiEigSym), so the rule is written down as not applying, with the reason: measures eigenvalue movement; the eigenvectors are never drawn

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 52 of 99 generators — 37 print a residual and 15 are exempt with a published reason; 47 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.

Eigenvalues, singular values, rank

A condition number for one eigenvalue

In the symmetric case every eigenvalue has condition number exactly one. In this four-by-four matrix two of them have condition number 100.005 and the other two have exactly 1, and the number belongs to the eigenvalue rather than to the matrix.

Iterating, instead of factorising

A rate that is known in advance

On the model problem, Jacobi contracts by cos(π/(n+1)) per step, Gauss–Seidel by its square, and optimally relaxed SOR by a number given in closed form. Three rates, all known before anything runs, and all measurable against what runs.

Orthogonality, measured

Orthogonal is a number

"Q is orthogonal" is a claim about a measurable quantity, ‖QᵀQ − I‖, and on the eight-by-eight Hilbert matrix two standard algorithms return 10⁻¹⁵ and 1 for it. The one that returns 1 still reconstructs the matrix perfectly, which is why nothing warns you.

Eigenvalues, singular values, rank

Rank is a decision

A floating-point matrix does not have a rank. It has a spectrum of singular values, and somewhere in that spectrum is a place where the values stop being signal and start being noise. Deciding where is a judgement, and the evidence for it is a gap.

Eigenvalues, singular values, rank

Symmetry is worth more than precision

A symmetric matrix gives up its eigenvalues to full accuracy however ill-conditioned it is. An unsymmetric one can move them by the eighth root of a perturbation, so the rounding involved in merely storing the matrix shifts the spectrum by a hundredth.

Eigenvalues, singular values, rank

The algorithm the libraries actually run

Factorise, multiply the factors back in the other order, repeat. That description is complete and correct and produces something nobody would use — on a matrix with eigenvalues +1 and −1 it does not converge at all, and the subdiagonal entry does not move by so much as a rounding error.

Two errors, and whose fault they are

The condition number is an amplifier

κ is usually introduced as a definition and then quoted. It is a measurement: perturb the input by a known amount, look at how much the output moves, and the largest ratio you can find is the number.

Two errors, and whose fault they are

The exact answer to a nearby problem

A good algorithm does not give an approximate answer to your problem. It gives the exact answer to a problem very close to yours — and once that is the definition, a wrong result has two possible authors and they can be measured apart.

Eigenvalues, singular values, rank

The form a real matrix can reach

A real matrix with complex eigenvalues has no real triangular form, and the reason is one line — a real triangular matrix has a real diagonal, and a similarity does not move the spectrum. What it has instead is triangular except for one two-by-two block per conjugate pair, and the count is decided by the matrix rather than by where the iteration stopped.

Eigenvalues, singular values, rank

The form that makes it affordable

One Householder reduction, done once, turns every subsequent iteration of the eigenvalue algorithm from cubic to quadratic cost. It changes no answer at all, which is why it is easy to describe as an optimisation and wrong to.

Eigenvalues, singular values, rank

The 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°.

Iterating, instead of factorising

The spectrum that predicts nothing

For a symmetric matrix the eigenvalues govern how fast an iteration converges. Drop symmetry and they stop governing anything — there is a matrix whose eigenvalues are as evenly spread as eigenvalues can be, on which GMRES makes no progress at all until the last possible step.

Orthogonality, measured

Two Gram–Schmidts

One argument changes. Classical Gram–Schmidt projects the original column onto each previous direction; modified projects what is left of it. In exact arithmetic the coefficients are identical. In floating point they differ by eight orders of magnitude in the thing that matters.

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