Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

GMRES on the Laplacian and on the cyclic shift, both 12×12

One function in the iterative library, called 7 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 32 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 gmres on the laplacian and on the cyclic shift, both 12×12. A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.

gmres-spectrum is one function in lib/figures/iterative.js — iterative — krylov and stationary methods against rates known in closed form. 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.

GMRES on the Laplacian and on the cyclic shift, both 12×12A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.02468101210⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1iteration‖r‖ / ‖b‖Laplaciancyclic shiftno progress at allevery eigenvalue of the shift is on the unit circleand it predicts nothing

A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.

size: 12

The arguments are the ones The spectrum that predicts nothing passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

GMRES on the Laplacian and on the cyclic shift, both 12×12A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.02468101210⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1iteration‖r‖ / ‖b‖Laplaciancyclic shiftno progress at allevery eigenvalue of the shift is on the unit circleand it predicts nothing

A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.

size: 6

The arguments are the ones The spectrum that predicts nothing passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

GMRES on the Laplacian and on the cyclic shift, both 6×6A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.024610⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1iteration‖r‖ / ‖b‖Laplaciancyclic shiftno progress at allevery eigenvalue of the shift is on the unit circleand it predicts nothing

A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.

size: 14

The arguments are the ones The spectrum that predicts nothing passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

GMRES on the Laplacian and on the cyclic shift, both 14×14A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.0246810121410⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1iteration‖r‖ / ‖b‖Laplaciancyclic shiftno progress at allevery eigenvalue of the shift is on the unit circleand it predicts nothing

A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.

size: 17

The arguments are the ones The spectrum that predicts nothing passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

GMRES on the Laplacian and on the cyclic shift, both 17×17A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.0369121510⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1iteration‖r‖ / ‖b‖Laplaciancyclic shiftno progress at allevery eigenvalue of the shift is on the unit circleand it predicts nothing

A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.

size: 20

The arguments are the ones The spectrum that predicts nothing passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

GMRES on the Laplacian and on the cyclic shift, both 20×20A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.036912151810⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1iteration‖r‖ / ‖b‖Laplaciancyclic shiftno progress at allevery eigenvalue of the shift is on the unit circleand it predicts nothing

A semi-logarithmic plot of relative residual against iteration. One curve falls steadily; the other is flat at one for every step until the last, where it drops to zero.

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.

32 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 shift has made no progress by step 1 — checked 19 times

the residual does not rise at step 1 — checked 10 times

and the shift is orthogonal, so every eigenvalue is on the unit circle

and then solves exactly at step n

matmul shapes agree

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.

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.

 Eigenvalues, singular values, rank

The vector was what was wanted

Nobody who computes a matrix exponential wants the matrix. They want eᴬᵗb — one vector, the state of a system at a later time. Twenty matrix–vector products get it to sixteen digits on a hundred-by-hundred problem, without ever forming a hundred-by-hundred exponential, and the exponential that does get computed is twenty by twenty.

The whole library · All essays · What must fail