Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

What 3 shifts do to each direction of the starting vector

One function in the restart library, called 13 times across 3 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 6 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 3 shifts do to each direction of the starting vector. Amplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 3 discarded Ritz values. They agree to 3.9·10⁻¹¹. The 3 wanted directions are amplified by at least 1 and everything else by at most 0.802.

restart-filter is one function in lib/figures/restart.js — restarting and blocks — a filter on the starting vector, and what one vector cannot see. 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 3 shifts do to each direction of the starting vectorAmplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 3 discarded Ritz values. They agree to 3.9·10⁻¹¹. The 3 wanted directions are amplified by at least 1 and everything else by at most 0.802.1357910⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1eigenvalue λamplificationkeptdiscardeda filter on the starting vectormeasured against the polynomial3.9·10⁻¹¹worst kept direction1best discarded direction0.8the roots are the discarded Ritz valuesand the vertical lines are where they sit

Amplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 3 discarded Ritz values. They agree to 3.9·10⁻¹¹. The 3 wanted directions are amplified by at least 1 and everything else by at most 0.802.

p: 3

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

What 3 shifts do to each direction of the starting vectorAmplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 3 discarded Ritz values. They agree to 3.9·10⁻¹¹. The 3 wanted directions are amplified by at least 1 and everything else by at most 0.802.1357910⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1eigenvalue λamplificationkeptdiscardeda filter on the starting vectormeasured against the polynomial3.9·10⁻¹¹worst kept direction1best discarded direction0.8the roots are the discarded Ritz valuesand the vertical lines are where they sit

Amplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 3 discarded Ritz values. They agree to 3.9·10⁻¹¹. The 3 wanted directions are amplified by at least 1 and everything else by at most 0.802.

p: 1

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

What 1 shift do to each direction of the starting vectorAmplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 1 discarded Ritz values. They agree to 2.2·10⁻¹³. The 3 wanted directions are amplified by at least 1.26 and everything else by at most 1.18.1357910⁻²10⁻¹1eigenvalue λamplificationkeptdiscardeda filter on the starting vectormeasured against the polynomial2.2·10⁻¹³worst kept direction1.3best discarded direction1.2the roots are the discarded Ritz valuesand the vertical lines are where they sit

Amplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 1 discarded Ritz values. They agree to 2.2·10⁻¹³. The 3 wanted directions are amplified by at least 1.26 and everything else by at most 1.18.

p: 6

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

What 6 shifts do to each direction of the starting vectorAmplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 6 discarded Ritz values. They agree to 2.8·10⁻⁹. The 3 wanted directions are amplified by at least 0.279 and everything else by at most 1.18·10⁻⁵.1357910⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1eigenvalue λamplificationkeptdiscardeda filter on the starting vectormeasured against the polynomial2.8·10⁻⁹worst kept direction0.28best discarded direction1.2·10⁻⁵the roots are the discarded Ritz valuesand the vertical lines are where they sit

Amplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 6 discarded Ritz values. They agree to 2.8·10⁻⁹. The 3 wanted directions are amplified by at least 0.279 and everything else by at most 1.18·10⁻⁵.

p: 4

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

What 4 shifts do to each direction of the starting vectorAmplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 4 discarded Ritz values. They agree to 3.3·10⁻¹¹. The 3 wanted directions are amplified by at least 0.803 and everything else by at most 0.548.1357910⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1eigenvalue λamplificationkeptdiscardeda filter on the starting vectormeasured against the polynomial3.3·10⁻¹¹worst kept direction0.8best discarded direction0.55the roots are the discarded Ritz valuesand the vertical lines are where they sit

Amplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 4 discarded Ritz values. They agree to 3.3·10⁻¹¹. The 3 wanted directions are amplified by at least 0.803 and everything else by at most 0.548.

p: 5

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

What 5 shifts do to each direction of the starting vectorAmplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 5 discarded Ritz values. They agree to 2.5·10⁻¹⁰. The 3 wanted directions are amplified by at least 0.32 and everything else by at most 0.00213.1357910⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1eigenvalue λamplificationkeptdiscardeda filter on the starting vectormeasured against the polynomial2.5·10⁻¹⁰worst kept direction0.32best discarded direction0.0021the roots are the discarded Ritz valuesand the vertical lines are where they sit

Amplification of each eigen-component against its eigenvalue, on a logarithmic vertical axis. The points are measured — the component of the vector along each eigenvector, before and after — and the curve is |Π(λ − θ)| with the roots at the 5 discarded Ritz values. They agree to 2.5·10⁻¹⁰. The 3 wanted directions are amplified by at least 0.32 and everything else by at most 0.00213.

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.

6 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 number of shifts a restart applies

and a keep count the basis has room for

and the wanted directions survive it best

every component is scaled by the same polynomial

Jacobi needs a symmetric matrix

matmul shapes agree

Against the rule

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

Restarting is a filter

A restart throws away the Ritz values it does not want and begins again from a new starting vector. Written in the eigenbasis, that vector's components have been multiplied by a polynomial with its roots at the discarded values — measured component by component, and agreeing with the polynomial to rounding.

 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 same budget, spent five ways

A restarted method has one budget — products with A — and two ways to spend it, in many short cycles or a few long ones. At about a hundred and forty products the answer is the same to a factor of seven whichever split is chosen, and the residual bound the method reports spans ten orders of magnitude across the same five runs.

The whole library · All essays · What must fail