Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

ichol-ageing

One function in the reuse library, called 6 times across 4 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 an incomplete cholesky kept while the operator turns anisotropic, against one rebuilt at every member. The five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.0331, with the sparsity pattern unchanged throughout — so the factorisation from the first member stays applicable for the whole run, which is the situation in which it gets kept. The kept one goes from 18 iterations to 52. The rebuilt one goes from 18 to 13, because an anisotropic operator is an easier problem for a factorisation that knows about the anisotropy. The two start at the same point by construction and never meet again.

ichol-ageing 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.

An incomplete Cholesky kept while the operator turns anisotropic, against one rebuilt at every memberThe five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.0331, with the sparsity pattern unchanged throughout — so the factorisation from the first member stays applicable for the whole run, which is the situation in which it gets kept. The kept one goes from 18 iterations to 52. The rebuilt one goes from 18 to 13, because an anisotropic operator is an easier problem for a factorisation that knows about the anisotropy. The two start at the same point by construction and never meet again.0246810121416182001020304050member of the sequencepreconditioned conjugate gradient iterationskept from the first memberrebuilt every memberone operator, two policieskept, first member18kept, last member52rebuilt, first member18rebuilt, last member13ε at the last member0.033the problem got easierand the kept preconditioner got worse at it

The five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.0331, with the sparsity pattern unchanged throughout — so the factorisation from the first member stays applicable for the whole run, which is the situation in which it gets kept. The kept one goes from 18 iterations to 52. The rebuilt one goes from 18 to 13, because an anisotropic operator is an easier problem for a factorisation that knows about the anisotropy. The two start at the same point by construction and never meet again.

k: 12

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.

An incomplete Cholesky kept while the operator turns anisotropic, against one rebuilt at every memberThe five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.0331, with the sparsity pattern unchanged throughout — so the factorisation from the first member stays applicable for the whole run, which is the situation in which it gets kept. The kept one goes from 18 iterations to 52. The rebuilt one goes from 18 to 13, because an anisotropic operator is an easier problem for a factorisation that knows about the anisotropy. The two start at the same point by construction and never meet again.0246810121416182001020304050member of the sequencepreconditioned conjugate gradient iterationskept from the first memberrebuilt every memberone operator, two policieskept, first member18kept, last member52rebuilt, first member18rebuilt, last member13ε at the last member0.033the problem got easierand the kept preconditioner got worse at it

The five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.0331, with the sparsity pattern unchanged throughout — so the factorisation from the first member stays applicable for the whole run, which is the situation in which it gets kept. The kept one goes from 18 iterations to 52. The rebuilt one goes from 18 to 13, because an anisotropic operator is an easier problem for a factorisation that knows about the anisotropy. The two start at the same point by construction and never meet again.

k: 14

The arguments are the ones What a rebuild is worth 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.

An incomplete Cholesky kept while the operator turns anisotropic, against one rebuilt at every memberThe five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.0331, with the sparsity pattern unchanged throughout — so the factorisation from the first member stays applicable for the whole run, which is the situation in which it gets kept. The kept one goes from 21 iterations to 54. The rebuilt one goes from 21 to 16, because an anisotropic operator is an easier problem for a factorisation that knows about the anisotropy. The two start at the same point by construction and never meet again.024681012141618200102030405060member of the sequencepreconditioned conjugate gradient iterationskept from the first memberrebuilt every memberone operator, two policieskept, first member21kept, last member54rebuilt, first member21rebuilt, last member16ε at the last member0.033the problem got easierand the kept preconditioner got worse at it

The five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.0331, with the sparsity pattern unchanged throughout — so the factorisation from the first member stays applicable for the whole run, which is the situation in which it gets kept. The kept one goes from 21 iterations to 54. The rebuilt one goes from 21 to 16, because an anisotropic operator is an easier problem for a factorisation that knows about the anisotropy. The two start at the same point by construction and never meet again.

k: 16

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.

An incomplete Cholesky kept while the operator turns anisotropic, against one rebuilt at every memberThe five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.0331, with the sparsity pattern unchanged throughout — so the factorisation from the first member stays applicable for the whole run, which is the situation in which it gets kept. The kept one goes from 23 iterations to 55. The rebuilt one goes from 23 to 17, because an anisotropic operator is an easier problem for a factorisation that knows about the anisotropy. The two start at the same point by construction and never meet again.024681012141618200102030405060member of the sequencepreconditioned conjugate gradient iterationskept from the first memberrebuilt every memberone operator, two policieskept, first member23kept, last member55rebuilt, first member23rebuilt, last member17ε at the last member0.033the problem got easierand the kept preconditioner got worse at it

The five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.0331, with the sparsity pattern unchanged throughout — so the factorisation from the first member stays applicable for the whole run, which is the situation in which it gets kept. The kept one goes from 23 iterations to 55. The rebuilt one goes from 23 to 17, because an anisotropic operator is an easier problem for a factorisation that knows about the anisotropy. The two start at the same point by construction and never meet again.

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 grid the dense incomplete factorisation is affordable on

and the two agree exactly at the member the factorisation was built from

every member converges on the kept factorisation

the kept factorisation more than doubles its iteration count over the sequence

while a rebuilt one takes fewer, because the problem itself got easier

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.

 Randomised, and the guarantee that changes kind

The sketch that is spent

Every other object a sequence carries has a shelf life. A random sketch has one use. Deflate what its first round found and apply it again, and it returns the zero matrix — 9.0·10⁻¹⁵ where the first round saw 2.62 — because the input has been made orthogonal to the very draw the guarantee is over.

 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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