Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Driving the subdiagonal to zero, with λ₄/λ₃ = 0.90

One function in the qralg 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 10 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 driving the subdiagonal to zero, with λ₄/λ₃ = 0.90. A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.

qr-iteration is one function in lib/figures/qralg.js — the qr algorithm — hessenberg, the shift, and what the libraries actually run. 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.

Driving the subdiagonal to zero, with λ₄/λ₃ = 0.90A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.0357010514017521024528010⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹iteration|subdiagonal entry|no shiftRayleighWilkinsontwo routes to one raterate, from the spectrum0.9rate, measured0.9iterations, none / Wilkinson45symmetric 4×4, spectrum 8, 4, 2, 1.8the dashed line is the prediction

A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.

gap: 0.9

The arguments are the ones The algorithm the libraries actually run passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Driving the subdiagonal to zero, with λ₄/λ₃ = 0.90A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.0357010514017521024528010⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹iteration|subdiagonal entry|no shiftRayleighWilkinsontwo routes to one raterate, from the spectrum0.9rate, measured0.9iterations, none / Wilkinson45symmetric 4×4, spectrum 8, 4, 2, 1.8the dashed line is the prediction

A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.

gap: 0.3

The arguments are the ones The form that makes it affordable passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Driving the subdiagonal to zero, with λ₄/λ₃ = 0.30A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.051015202530354010⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹iteration|subdiagonal entry|no shiftRayleighWilkinsontwo routes to one raterate, from the spectrum0.3rate, measured0.3iterations, none / Wilkinson6.7symmetric 4×4, spectrum 8, 4, 2, 0.6the dashed line is the prediction

A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.

gap: 0.5392857142857143

The arguments are the ones The form that makes it affordable passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Driving the subdiagonal to zero, with λ₄/λ₃ = 0.54A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.0816243240485610⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹iteration|subdiagonal entry|no shiftRayleighWilkinsontwo routes to one raterate, from the spectrum0.54rate, measured0.54iterations, none / Wilkinson7.1symmetric 4×4, spectrum 8, 4, 2, 1.08the dashed line is the prediction

A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.

gap: 0.8264285714285713

The arguments are the ones The form that makes it affordable passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Driving the subdiagonal to zero, with λ₄/λ₃ = 0.83A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.02142638410512614710⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹iteration|subdiagonal entry|no shiftRayleighWilkinsontwo routes to one raterate, from the spectrum0.83rate, measured0.83iterations, none / Wilkinson22symmetric 4×4, spectrum 8, 4, 2, 1.65the dashed line is the prediction

A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.

gap: 0.9221428571428572

The arguments are the ones The form that makes it affordable passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Driving the subdiagonal to zero, with λ₄/λ₃ = 0.92A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.0459013518022527031510⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹iteration|subdiagonal entry|no shiftRayleighWilkinsontwo routes to one raterate, from the spectrum0.92rate, measured0.92iterations, none / Wilkinson58symmetric 4×4, spectrum 8, 4, 2, 1.84the dashed line is the prediction

A semi-logarithmic plot of the magnitude of the subdiagonal entry against iteration count for three shift strategies. The unshifted curve is a straight line; the two shifted curves plunge to the bottom of the plot within a few steps.

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.

10 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.

Wilkinson returns eigenvalue 1 — checked 4 times

and so does Rayleigh

in fewer iterations than the unshifted one

matmul shapes agree

the unshifted algorithm contracts at the ratio of the two eigenvalues

the unshifted run has a rate to measure

the Wilkinson-shifted algorithm converged

Against the rule

It draws a decomposition and prints its residual. It calls qrAlgorithm, 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.

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.

 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

Two matrices and one problem

Ax = λBx is what a finite element model, a structural vibration and a constrained optimisation actually produce, and it is not the one-matrix problem with a change of variables. Everybody is told not to form B⁻¹A because it is not symmetric. That is true, the departure from symmetry is about one, and it is not what decides the accuracy.

The whole library · All essays · What must fail