Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

A 4-level hierarchy on 15 points, and the coarse operator it implies

One function in the multigrid library, called 3 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 7 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 a 4-level hierarchy on 15 points, and the coarse operator it implies. Rows of dots, each row half the length of the one above it, with lines joining every coarse point to the three fine points it interpolates to.

grid-hierarchy is one function in lib/figures/multigrid.js — multigrid — the smoother, the coarse grid, and the rate that does not move. 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.

A 4-level hierarchy on 15 points, and the coarse operator it impliesRows of dots, each row half the length of the one above it, with lines joining every coarse point to the three fine points it interpolates to.15731pointsfinest gridone unknown — the recursion bottoms out in a divisionthe coarse operator, two ways‖RAₕP − A₂ₕ‖/‖A₂ₕ‖10⁻¹⁸unknowns / finest grid1.7cycle cost, in fine sweeps12each coarse point reaches three fine ones½, 1, ½ — and the restriction is its transpose

Rows of dots, each row half the length of the one above it, with lines joining every coarse point to the three fine points it interpolates to.

levels: 6

The arguments are the ones A rate that does not notice the size passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

A 6-level hierarchy on 63 points, and the coarse operator it impliesRows of dots, each row half the length of the one above it, with lines joining every coarse point to the three fine points it interpolates to.633115731pointsfinest gridone unknown — the recursion bottoms out in a divisionthe coarse operator, two ways‖RAₕP − A₂ₕ‖/‖A₂ₕ‖10⁻¹⁸unknowns / finest grid1.9cycle cost, in fine sweeps14each coarse point reaches three fine ones½, 1, ½ — and the restriction is its transpose

Rows of dots, each row half the length of the one above it, with lines joining every coarse point to the three fine points it interpolates to.

levels: 4

The arguments are the ones The same problem on a coarser grid passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

A 4-level hierarchy on 15 points, and the coarse operator it impliesRows of dots, each row half the length of the one above it, with lines joining every coarse point to the three fine points it interpolates to.15731pointsfinest gridone unknown — the recursion bottoms out in a divisionthe coarse operator, two ways‖RAₕP − A₂ₕ‖/‖A₂ₕ‖10⁻¹⁸unknowns / finest grid1.7cycle cost, in fine sweeps12each coarse point reaches three fine ones½, 1, ½ — and the restriction is its transpose

Rows of dots, each row half the length of the one above it, with lines joining every coarse point to the three fine points it interpolates to.

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.

7 distinct claims across 3 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 Galerkin operator at level 1 is the coarse discretisation — checked 5 times

matmul shapes agree

the hierarchy has at least three levels and at most six

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

A rate that does not notice the size

The V-cycle reduces the residual by a factor of ten a cycle at fifteen points and at a hundred and twenty-seven. Jacobi on the same four problems goes from 0.981 to 0.9978, climbing towards one. One of those is a constant and the other is an exponent, and that is the whole distinction the field turns on.

 Iterating, instead of factorising

The same problem on a coarser grid

Restriction, the coarse operator and interpolation are three matrices with nine distinct entries between them. Two of the three are each other's transpose, and their product with the fine operator is the coarse discretisation exactly — not approximately, entry for entry, at every level.

The whole library · All essays · What must fail