Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Smoothing factor against anisotropy, at ω = 0.667

One function in the grid2d library, called 8 times across 3 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 4 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 smoothing factor against anisotropy, at ω = 0.667. Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.

anisotropy-factors is one function in lib/figures/grid2d.js — two dimensions — where the galerkin identity stops being an identity. 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.

Smoothing factor against anisotropy, at ω = 0.667Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.10⁻⁴10⁻³10⁻²10⁻¹100.250.50.751anisotropy εsmoothing factor μpoint + fully-line + fullpoint + semi-y⅓, the one-dimensional answertwo routes, three curvesgap between the repairs0scan against closed form1.1·10⁻¹⁶the dashed curve lies on the solid one beneath itone repair, written two ways

Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.

omega: 0.6667

The arguments are the ones A direction the smoother cannot see passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Smoothing factor against anisotropy, at ω = 0.667Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.10⁻⁴10⁻³10⁻²10⁻¹100.250.50.751anisotropy εsmoothing factor μpoint + fully-line + fullpoint + semi-y⅓, the one-dimensional answertwo routes, three curvesgap between the repairs2.2·10⁻¹⁶scan against closed form2.2·10⁻¹⁶the dashed curve lies on the solid one beneath itone repair, written two ways

Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.

omega: 0.8

The arguments are the ones How much direction there was to lose passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Smoothing factor against anisotropy, at ω = 0.800Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.10⁻⁴10⁻³10⁻²10⁻¹100.250.50.751anisotropy εsmoothing factor μpoint + fully-line + fullpoint + semi-y⅓, the one-dimensional answertwo routes, three curvesgap between the repairs3.7·10⁻¹³scan against closed form3.7·10⁻¹³the dashed curve lies on the solid one beneath itone repair, written two ways

Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.

omega: 0.5

The arguments are the ones Smoothing a whole line at once passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Smoothing factor against anisotropy, at ω = 0.500Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.10⁻⁴10⁻³10⁻²10⁻¹100.250.50.751anisotropy εsmoothing factor μpoint + fully-line + fullpoint + semi-y⅓, the one-dimensional answertwo routes, three curvesgap between the repairs0scan against closed form1.1·10⁻¹⁶the dashed curve lies on the solid one beneath itone repair, written two ways

Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.

omega: 1

The arguments are the ones Smoothing a whole line at once passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Smoothing factor against anisotropy, at ω = 1.000Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.10⁻⁴10⁻³10⁻²10⁻¹100.250.50.751anisotropy εsmoothing factor μpoint + fully-line + fullpoint + semi-y⅓, the one-dimensional answertwo routes, three curvesgap between the repairs4.7·10⁻¹³scan against closed form4.7·10⁻¹³the dashed curve lies on the solid one beneath itone repair, written two ways

Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.

omega: 0.75

The arguments are the ones Smoothing a whole line at once passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Smoothing factor against anisotropy, at ω = 0.750Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.10⁻⁴10⁻³10⁻²10⁻¹100.250.50.751anisotropy εsmoothing factor μpoint + fully-line + fullpoint + semi-y⅓, the one-dimensional answertwo routes, three curvesgap between the repairs3.5·10⁻¹³scan against closed form3.5·10⁻¹³the dashed curve lies on the solid one beneath itone repair, written two ways

Three curves of the smoothing factor against the anisotropy parameter on a logarithmic axis. One rises to one as the anisotropy grows; the other two coincide and stay near a third.

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.

4 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.

and every scanned value is its closed form

and near the best weight it is far worse

at the smallest ε the point smoother is the worse of the two at every weight

the two repairs have the same smoothing factor at every ε drawn

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 direction the smoother cannot see

Give the Laplacian a strong direction and multigrid stops working — from 0.2016 a cycle to 0.9565 — with every component unchanged and the condition number identical to twelve digits. The problem did not get harder. The link between the method's two halves broke.

 Iterating, instead of factorising

How much direction there was to lose

At 45° the nine-point stencil hands smoothed aggregation the same wrong hierarchy at every anisotropy — six strong neighbours per interior point, 121 aggregates, the identical partition from ε = 10⁻⁴ to 0.099. The convergence factor that one hierarchy produces runs from 0.802 to 0.581 over the same range.

 Iterating, instead of factorising

Smoothing a whole line at once

Solve every grid line in the strong direction exactly rather than sweeping over it, and the smoothing factor goes from 0.9993 back to 0.3340 — which is the one-dimensional answer, on a problem that is not one-dimensional. The repair replaces one ε in the closed form by a one.

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