Two-level convergence factor for three interpolations, at ε = 0.001
At its defaults it draws two-level convergence factor for three interpolations, at ε = 0.001. Three bars per operator: the classical formula, the unconstrained energy minimiser, and the minimiser constrained to reproduce a constant. On the isotropic and aligned operators the constraint costs a factor of 4.8; on the rotated one it buys a factor of 1.33.
constraint-cost is one function in lib/figures/emin.js —
interpolation as a minimisation — where the classical formula already was the answer. 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.
Three bars per operator: the classical formula, the unconstrained energy minimiser, and the minimiser constrained to reproduce a constant. On the isotropic and aligned operators the constraint costs a factor of 4.8; on the rotated one it buys a factor of 1.33.
eps: 0.001
The arguments are the ones The formula that was already optimal passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three bars per operator: the classical formula, the unconstrained energy minimiser, and the minimiser constrained to reproduce a constant. On the isotropic and aligned operators the constraint costs a factor of 4.8; on the rotated one it buys a factor of 1.33.
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 2 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 comparison is affordable at
an anisotropy strong enough for the three interpolations to differ
and buys it on the rotated one
LU is for square matrices
the constraint costs convergence on the aligned operator
the constraint costs convergence on the isotropic operator
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.
The formula that was already optimal
Ask for the interpolation that minimises the energy of its own columns and the answer is the classical AMG formula — to zero at every row of the one-dimensional Laplacian, and to four digits in two dimensions. On the operator rotated to 45° the two part company, and the gap between them is a diagnostic that needs no reference solution.
The matrix a constraint makesTwo ways to remove a constraint
A constrained system can be reduced by eliminating the multipliers or by eliminating the constrained directions. Both give the same answer in exact arithmetic and inherit different condition numbers — one of them squares the constraint's, and the other does not contain it at all.