Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

breakdown-onset

One function in the breakdown library, called 5 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 5 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 the step at which the arnoldi recurrence breaks down, against the number of distinct eigenvalues, n = 30. Every matrix here is 30×30 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 30, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 5.64·10⁻¹⁶; one step earlier it is at least 3.5·10¹⁰ times larger.

breakdown-onset is one function in lib/figures/breakdown.js — breakdown — the zero that is an answer, and the zero that is nothing. 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.

The step at which the Arnoldi recurrence breaks down, against the number of distinct eigenvalues, n = 30Every matrix here is 30×30 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 30, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 5.64·10⁻¹⁶; one step earlier it is at least 3.5·10¹⁰ times larger.024681012141618024681012141618distinct eigenvalues in the spectrumstep the recurrence stops atthe step is m, not nn = 30 throughoutspectra drawn8every one breaking at m8worst residual at the breakdown5.6·10⁻¹⁶smallest gain over the step before3.5·10¹⁰an invariant subspace contains the answerand its dimension is what the method costs

Every matrix here is 30×30 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 30, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 5.64·10⁻¹⁶; one step earlier it is at least 3.5·10¹⁰ times larger.

n: 30

The arguments are the ones Deciding that a zero has arrived 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.

The step at which the Arnoldi recurrence breaks down, against the number of distinct eigenvalues, n = 30Every matrix here is 30×30 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 30, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 5.64·10⁻¹⁶; one step earlier it is at least 3.5·10¹⁰ times larger.024681012141618024681012141618distinct eigenvalues in the spectrumstep the recurrence stops atthe step is m, not nn = 30 throughoutspectra drawn8every one breaking at m8worst residual at the breakdown5.6·10⁻¹⁶smallest gain over the step before3.5·10¹⁰an invariant subspace contains the answerand its dimension is what the method costs

Every matrix here is 30×30 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 30, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 5.64·10⁻¹⁶; one step earlier it is at least 3.5·10¹⁰ times larger.

n: 60

The arguments are the ones The zero that means it is finished 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.

The step at which the Arnoldi recurrence breaks down, against the number of distinct eigenvalues, n = 60Every matrix here is 60×60 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 60, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 8.34·10⁻¹⁶; one step earlier it is at least 1.6·10¹⁰ times larger.024681012141618024681012141618distinct eigenvalues in the spectrumstep the recurrence stops atthe step is m, not nn = 60 throughoutspectra drawn8every one breaking at m8worst residual at the breakdown8.3·10⁻¹⁶smallest gain over the step before1.6·10¹⁰an invariant subspace contains the answerand its dimension is what the method costs

Every matrix here is 60×60 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 60, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 8.34·10⁻¹⁶; one step earlier it is at least 1.6·10¹⁰ times larger.

n: 45

The arguments are the ones The zero that means it is finished 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.

The step at which the Arnoldi recurrence breaks down, against the number of distinct eigenvalues, n = 45Every matrix here is 45×45 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 45, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 9.65·10⁻¹⁶; one step earlier it is at least 1.7·10¹⁰ times larger.024681012141618024681012141618distinct eigenvalues in the spectrumstep the recurrence stops atthe step is m, not nn = 45 throughoutspectra drawn8every one breaking at m8worst residual at the breakdown9.6·10⁻¹⁶smallest gain over the step before1.7·10¹⁰an invariant subspace contains the answerand its dimension is what the method costs

Every matrix here is 45×45 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 45, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 9.65·10⁻¹⁶; one step earlier it is at least 1.7·10¹⁰ times larger.

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.

5 distinct claims across 4 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 matrix size the sweep can run every spectrum inside

an Arnoldi recurrence started from a vector that is not zero

and the answer there is exact

enough distinct spectra to make the line a line

the recurrence breaks down at the number of distinct eigenvalues

Against the rule

It draws a decomposition and prints its residual. It calls onsetSweep, 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 113 of 219 generators — 98 print a residual and 15 are exempt with a published reason; 106 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.

Two errors, and whose fault they are

Deciding that a zero has arrived

The previous tolerances were offers — accept this much error, save this much work. A detection threshold is not an offer, because both directions are failures. One matrix here has three genuinely near-invariant subspaces, and the constant somebody typed decides which of them the recurrence stops at; at eight significand bits the same kind of constant produces a proof of something false.

 Iterating, instead of factorising

The zero that means it is finished

Every Krylov method ends by dividing by a number the previous step produced, and when that number is zero the recurrence stops. In Arnoldi the stop is the answer — the subspace has closed, the solution is inside it, and the residual is at the unit roundoff. The literature calls it a lucky breakdown, and the adjective is doing real work.

The whole library · All essays · What must fail