Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

What a perturbation of 10⁻⁶ moves, against the gap it is applied across

One function in the subspace library, called 8 times across 3 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 8 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 what a perturbation of 10⁻⁶ moves, against the gap it is applied across. Three curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.

gap-sweep is one function in lib/figures/subspace.js — subspaces — the gap, the eigenvector, and the plane that survives both. 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.

What a perturbation of 10⁻⁶ moves, against the gap it is applied acrossThree curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻⁸10⁻⁵10⁻²gap between the two eigenvalueshow far it movedthe eigenvectorsthe eigenvaluestheir plane‖E‖ / gapone perturbation, three answerseigenvalue shift, spread over the sweep1plane angle, spread over the sweep1eigenvector angle, spread1.6·10⁵the dashed line is Davis–Kahan's ‖E‖/gaptwo of the three never noticed

Three curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.

size: 0.000001

The arguments are the ones The gap decides the eigenvector passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What a perturbation of 10⁻⁶ moves, against the gap it is applied acrossThree curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻⁸10⁻⁵10⁻²gap between the two eigenvalueshow far it movedthe eigenvectorsthe eigenvaluestheir plane‖E‖ / gapone perturbation, three answerseigenvalue shift, spread over the sweep1plane angle, spread over the sweep1eigenvector angle, spread1.6·10⁵the dashed line is Davis–Kahan's ‖E‖/gaptwo of the three never noticed

Three curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.

size: 1e-10

The arguments are the ones The gap decides the eigenvector passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What a perturbation of 10⁻¹⁰ moves, against the gap it is applied acrossThree curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴gap between the two eigenvalueshow far it movedthe eigenvectorsthe eigenvaluestheir plane‖E‖ / gapone perturbation, three answerseigenvalue shift, spread over the sweep1plane angle, spread over the sweep1eigenvector angle, spread∞the dashed line is Davis–Kahan's ‖E‖/gaptwo of the three never noticed

Three curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.

size: 1e-8

The arguments are the ones The gap decides the eigenvector passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What a perturbation of 10⁻⁸ moves, against the gap it is applied acrossThree curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹gap between the two eigenvalueshow far it movedthe eigenvectorsthe eigenvaluestheir plane‖E‖ / gapone perturbation, three answerseigenvalue shift, spread over the sweep1plane angle, spread over the sweep1eigenvector angle, spread5.6·10⁶the dashed line is Davis–Kahan's ‖E‖/gaptwo of the three never noticed

Three curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.

size: 0.00001

The arguments are the ones The gap decides the eigenvector passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What a perturbation of 10⁻⁵ moves, against the gap it is applied acrossThree curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻⁷10⁻⁴10⁻¹gap between the two eigenvalueshow far it movedthe eigenvectorsthe eigenvaluestheir plane‖E‖ / gapone perturbation, three answerseigenvalue shift, spread over the sweep1plane angle, spread over the sweep1eigenvector angle, spread1.6·10⁴the dashed line is Davis–Kahan's ‖E‖/gaptwo of the three never noticed

Three curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.

size: 0.0001

The arguments are the ones The gap decides the eigenvector passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What a perturbation of 10⁻⁴ moves, against the gap it is applied acrossThree curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻⁶10⁻³1gap between the two eigenvalueshow far it movedthe eigenvectorsthe eigenvaluestheir plane‖E‖ / gapone perturbation, three answerseigenvalue shift, spread over the sweep1plane angle, spread over the sweep1eigenvector angle, spread1583the dashed line is Davis–Kahan's ‖E‖/gaptwo of the three never noticed

Three curves against the gap between two eigenvalues, both axes logarithmic. Two are flat; the third rises steeply as the gap closes and then levels off.

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.

8 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 small enough to be a perturbation

and neither does the plane's angle

and the plane stays inside its own bound

Jacobi needs a symmetric matrix

matmul shapes agree

no eigenvalue moves by more than the perturbation

the eigenvalue shift does not move with the gap

while the eigenvector's angle moves by orders of magnitude

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 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

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°.

 Eigenvalues, singular values, rank

The 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.

 Least squares, and the road not to take

The valley with no bottom

A degree-nine fit's coefficients can be moved by a third of their own size before the residual changes in the sixth significant figure. The arithmetic did not lose those digits. The data never contained them.

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