Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

drift-ends

One function in the reuse library, called 9 times across 4 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 20 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 preconditioned iterations against the size of the drift, for a drift on each half of the spectrum. The preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 38 and 9 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 10⁴.

drift-ends is one function in lib/figures/reuse.js — reuse — what a kept factorisation or preconditioner is worth, and for how long. 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.

Preconditioned iterations against the size of the drift, for a drift on each half of the spectrumThe preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 38 and 9 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 10⁴.10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10112233445566relative drift ‖E‖ / ‖A₀‖preconditioned conjugate gradient iterationson the small eigenvalueson the large onessame drift, two placessmall end at 10⁻²38large end at 10⁻²9κ(M⁻¹A), small end36κ(M⁻¹A), large end1.3‖E‖/‖A₀‖, both0.01how much the matrix changedis not what the preconditioner cares about

The preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 38 and 9 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 10⁴.

logKappa: 4

The arguments are the ones A factorisation kept past its date 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.

Preconditioned iterations against the size of the drift, for a drift on each half of the spectrumThe preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 38 and 9 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 10⁴.10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10112233445566relative drift ‖E‖ / ‖A₀‖preconditioned conjugate gradient iterationson the small eigenvalueson the large onessame drift, two placessmall end at 10⁻²38large end at 10⁻²9κ(M⁻¹A), small end36κ(M⁻¹A), large end1.3‖E‖/‖A₀‖, both0.01how much the matrix changedis not what the preconditioner cares about

The preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 38 and 9 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 10⁴.

logKappa: 6

The arguments are the ones Where the drift lands 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.

Preconditioned iterations against the size of the drift, for a drift on each half of the spectrumThe preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 82 and 16 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 10⁶.10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹101632486480relative drift ‖E‖ / ‖A₀‖preconditioned conjugate gradient iterationson the small eigenvalueson the large onessame drift, two placessmall end at 10⁻²82large end at 10⁻²16κ(M⁻¹A), small end2911κ(M⁻¹A), large end3.7‖E‖/‖A₀‖, both0.01how much the matrix changedis not what the preconditioner cares about

The preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 82 and 16 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 10⁶.

logKappa: 2

The arguments are the ones Where the drift lands 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.

Preconditioned iterations against the size of the drift, for a drift on each half of the spectrumThe preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 10 and 6 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 100.10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1061218243036relative drift ‖E‖ / ‖A₀‖preconditioned conjugate gradient iterationson the small eigenvalueson the large onessame drift, two placessmall end at 10⁻²10large end at 10⁻²6κ(M⁻¹A), small end1.5κ(M⁻¹A), large end1‖E‖/‖A₀‖, both0.01how much the matrix changedis not what the preconditioner cares about

The preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 10 and 6 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 100.

logKappa: 3

The arguments are the ones Where the drift lands 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.

Preconditioned iterations against the size of the drift, for a drift on each half of the spectrumThe preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 21 and 7 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 1000.10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10918273645relative drift ‖E‖ / ‖A₀‖preconditioned conjugate gradient iterationson the small eigenvalueson the large onessame drift, two placessmall end at 10⁻²21large end at 10⁻²7κ(M⁻¹A), small end4.9κ(M⁻¹A), large end1.1‖E‖/‖A₀‖, both0.01how much the matrix changedis not what the preconditioner cares about

The preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 21 and 7 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 1000.

logKappa: 5

The arguments are the ones Where the drift lands 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.

Preconditioned iterations against the size of the drift, for a drift on each half of the spectrumThe preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 64 and 12 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 10⁵.10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10132639526578relative drift ‖E‖ / ‖A₀‖preconditioned conjugate gradient iterationson the small eigenvalueson the large onessame drift, two placessmall end at 10⁻²64large end at 10⁻²12κ(M⁻¹A), small end315κ(M⁻¹A), large end1.9‖E‖/‖A₀‖, both0.01how much the matrix changedis not what the preconditioner cares about

The preconditioner is the exact Cholesky factorisation of the first matrix, so it solves that matrix in one step and every iteration counted here is bought by the drift. Both curves are drifts of the same relative Frobenius norm; the upper one sits on the half of the spectrum with the small eigenvalues and the lower on the half with the large. At a relative drift of 10⁻² they cost 64 and 12 iterations, because the drift is divided by the eigenvalue it lands on: the preconditioned matrix has eigenvalues 1 + w/λ, and λ runs over 10⁵.

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.

20 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 two drifts at 0.001 have the same relative Frobenius norm — asserted 4 times

and at a relative drift of 0.001 the small end never costs less than the large one — asserted 3 times

a conditioning the construction holds exactly

a half of the spectrum to place the drift in

a relative drift inside the range the figures use

a size the dense eigendecomposition is affordable at

and the gap between them grows with the condition number, which is the quantity the norm cannot see

Jacobi needs a symmetric matrix

matmul shapes agree

the preconditioner solves its own matrix in one step, so the sweep measures the drift alone

the same drifts at both ends

the spectrum spans the condition number it was asked for

the two drifts at 10⁻⁴ have the same relative Frobenius norm

the two drifts at 10⁻⁵ have the same relative Frobenius norm

the two drifts at 10⁻⁶ have the same relative Frobenius norm

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 113 of 238 generators — 98 print a residual and 15 are exempt with a published reason; 125 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.

When the problem arrives again

A factorisation kept past its date

One Cholesky factor can serve five members of a drifting sequence and save 44 per cent of the work. Kept for twenty it does not lose accuracy — it stops converging altogether. The optimum and the cliff are four members apart, both move with the drift, and a rule written in a ratio the iteration has already computed finds them without being told what the drift is.

 When the problem arrives again

The accuracy that is thrown away

A Newton step is the exact answer to a linearised problem, and the linearisation is wrong at second order. So there is a floor under how close the step can land, the floor is the square of where it started, and eleven decades of inner tolerance below it buy the same four digits at four times the price.

 When the problem arrives again

What a rebuild is worth

One sequence, one drift, one preconditioner — and six different right answers, because the cheapest rebuild period depends on what a rebuild cost to build and on nothing else. The optimum walks from every member to every twelfth as the setup gets dearer, and the free rule that reads the iteration count beats it in the middle of that range and loses at both ends.

 When the problem arrives again

Where the drift lands

The standing rule for when a preconditioner has gone stale is to rebuild it once the matrix has changed by more than some fraction of itself. Two drifts of exactly the same relative size cost 19 iterations and 5 on the same matrix, and the quantity that separates them is not in the rule at all — the perturbation is divided by the eigenvalue it lands on.

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