Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

How far a perturbation of size ε moves an eigenvalue, 8×8

One function in the spectra library, called 6 times across 1 essay. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 41 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 1/8, and at a perturbation of ten to the minus sixteen it has already moved by 0.01.

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.

How far a perturbation of size ε moves an eigenvalue, 8×8A 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 1/8, and at a perturbation of ten to the minus sixteen it has already moved by 0.01.10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹size of the perturbation ‖δA‖how far the eigenvalues moveJordan block, ε^(1/8)symmetric, ≤ ‖δA‖rounding error alone moves it to 10⁻²six seeds per symmetric point; Jordan is closed formsymmetry beats precision

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 1/8, and at a perturbation of ten to the minus sixteen it has already moved by 0.01.

n: 8

The arguments are the ones Symmetry is worth more than precision passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

How far a perturbation of size ε moves an eigenvalue, 8×8A 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 1/8, and at a perturbation of ten to the minus sixteen it has already moved by 0.01.10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹size of the perturbation ‖δA‖how far the eigenvalues moveJordan block, ε^(1/8)symmetric, ≤ ‖δA‖rounding error alone moves it to 10⁻²six seeds per symmetric point; Jordan is closed formsymmetry beats precision

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 1/8, and at a perturbation of ten to the minus sixteen it has already moved by 0.01.

n: 6

The arguments are the ones Symmetry is worth more than precision passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

How far a perturbation of size ε moves an eigenvalue, 6×6A 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 1/6, and at a perturbation of ten to the minus sixteen it has already moved by 0.00215.10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹size of the perturbation ‖δA‖how far the eigenvalues moveJordan block, ε^(1/6)symmetric, ≤ ‖δA‖rounding error alone moves it to 10⁻²six seeds per symmetric point; Jordan is closed formsymmetry beats precision

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 1/6, and at a perturbation of ten to the minus sixteen it has already moved by 0.00215.

n: 7

The arguments are the ones Symmetry is worth more than precision passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

How far a perturbation of size ε moves an eigenvalue, 7×7A 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 1/7, and at a perturbation of ten to the minus sixteen it has already moved by 0.00518.10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹size of the perturbation ‖δA‖how far the eigenvalues moveJordan block, ε^(1/7)symmetric, ≤ ‖δA‖rounding error alone moves it to 10⁻²six seeds per symmetric point; Jordan is closed formsymmetry beats precision

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 1/7, and at a perturbation of ten to the minus sixteen it has already moved by 0.00518.

n: 9

The arguments are the ones Symmetry is worth more than precision passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

How far a perturbation of size ε moves an eigenvalue, 9×9A 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 1/9, and at a perturbation of ten to the minus sixteen it has already moved by 0.0167.10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹size of the perturbation ‖δA‖how far the eigenvalues moveJordan block, ε^(1/9)symmetric, ≤ ‖δA‖rounding error alone moves it to 10⁻²six seeds per symmetric point; Jordan is closed formsymmetry beats precision

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 1/9, and at a perturbation of ten to the minus sixteen it has already moved by 0.0167.

n: 10

The arguments are the ones Symmetry is worth more than precision passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

How far a perturbation of size ε moves an eigenvalue, 10×10A 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 1/10, and at a perturbation of ten to the minus sixteen it has already moved by 0.0251.10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹size of the perturbation ‖δA‖how far the eigenvalues moveJordan block, ε^(1/10)symmetric, ≤ ‖δA‖rounding error alone moves it to 10⁻²six seeds per symmetric point; Jordan is closed formsymmetry beats precision

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 1/10, and at a perturbation of ten to the minus sixteen it has already moved by 0.0251.

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.

41 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.

the 8-th root of ε is an eigenvalue of the perturbed Jordan block — checked 5 times

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

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

a size the cofactor verification of the characteristic polynomial can afford

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

Jacobi needs a symmetric matrix

matmul shapes agree

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 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.

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.

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