Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

The same preconditioner in one dimension and in two (ρ = 0.9)

One function in the bttb library, called 6 times across 1 essay. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 11 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 the same preconditioner in one dimension and in two (ρ = 0.9). Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 7, 10, 10, 10 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 20, 21, climbing, against an unpreconditioned 14, 37, 73, 115. Nothing about the construction changed.

dimension-steps is one function in lib/figures/bttb.js — block toeplitz — the second dimension, where the cluster thins. 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.

The same preconditioner in one dimension and in two (ρ = 0.9)Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 7, 10, 10, 10 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 20, 21, climbing, against an unpreconditioned 14, 37, 73, 115. Nothing about the construction changed.10²020406080100120unknownsiterations2D, no preconditioner2D, preconditioned1D, preconditionedone construction, two dimensions2D steps at 16 unknowns102D steps at 100 unknowns211D steps at 100 unknowns10the same averaging, the same transformand a count that no longer stops growing

Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 7, 10, 10, 10 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 20, 21, climbing, against an unpreconditioned 14, 37, 73, 115. Nothing about the construction changed.

rho: 0.9

The arguments are the ones Two dimensions, and the cluster that thins passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The same preconditioner in one dimension and in two (ρ = 0.9)Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 7, 10, 10, 10 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 20, 21, climbing, against an unpreconditioned 14, 37, 73, 115. Nothing about the construction changed.10²020406080100120unknownsiterations2D, no preconditioner2D, preconditioned1D, preconditionedone construction, two dimensions2D steps at 16 unknowns102D steps at 100 unknowns211D steps at 100 unknowns10the same averaging, the same transformand a count that no longer stops growing

Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 7, 10, 10, 10 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 20, 21, climbing, against an unpreconditioned 14, 37, 73, 115. Nothing about the construction changed.

rho: 0.5

The arguments are the ones Two dimensions, and the cluster that thins passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The same preconditioner in one dimension and in two (ρ = 0.5)Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 8, 8, 7, 7 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 17, 17, 18, climbing, against an unpreconditioned 10, 23, 37, 43. Nothing about the construction changed.10²02040unknownsiterations2D, no preconditioner2D, preconditioned1D, preconditionedone construction, two dimensions2D steps at 16 unknowns102D steps at 100 unknowns181D steps at 100 unknowns7the same averaging, the same transformand a count that no longer stops growing

Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 8, 8, 7, 7 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 17, 17, 18, climbing, against an unpreconditioned 10, 23, 37, 43. Nothing about the construction changed.

rho: 0.7

The arguments are the ones Two dimensions, and the cluster that thins passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The same preconditioner in one dimension and in two (ρ = 0.7)Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 8, 9, 8, 7 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 20, 21, climbing, against an unpreconditioned 12, 28, 50, 63. Nothing about the construction changed.10²0204060unknownsiterations2D, no preconditioner2D, preconditioned1D, preconditionedone construction, two dimensions2D steps at 16 unknowns102D steps at 100 unknowns211D steps at 100 unknowns7the same averaging, the same transformand a count that no longer stops growing

Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 8, 9, 8, 7 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 20, 21, climbing, against an unpreconditioned 12, 28, 50, 63. Nothing about the construction changed.

rho: 0.8

The arguments are the ones Two dimensions, and the cluster that thins passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The same preconditioner in one dimension and in two (ρ = 0.8)Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 8, 9, 8, 8 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 22, 21, climbing, against an unpreconditioned 12, 31, 60, 82. Nothing about the construction changed.10²020406080unknownsiterations2D, no preconditioner2D, preconditioned1D, preconditionedone construction, two dimensions2D steps at 16 unknowns102D steps at 100 unknowns211D steps at 100 unknowns8the same averaging, the same transformand a count that no longer stops growing

Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 8, 9, 8, 8 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 22, 21, climbing, against an unpreconditioned 12, 31, 60, 82. Nothing about the construction changed.

rho: 0.95

The arguments are the ones Two dimensions, and the cluster that thins passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The same preconditioner in one dimension and in two (ρ = 0.95)Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 8, 9, 9, 10 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 19, 19, climbing, against an unpreconditioned 15, 41, 90, 143. Nothing about the construction changed.10²020406080100120140160unknownsiterations2D, no preconditioner2D, preconditioned1D, preconditionedone construction, two dimensions2D steps at 16 unknowns102D steps at 100 unknowns191D steps at 100 unknowns10the same averaging, the same transformand a count that no longer stops growing

Iteration count against the number of unknowns, on a logarithmic size axis. In one dimension the preconditioned count is 8, 9, 9, 10 — flat across a factor of six in size. On square grids with the same unknown counts it is 10, 18, 19, 19, climbing, against an unpreconditioned 15, 41, 90, 143. Nothing about the construction changed.

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.

11 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 preconditioner does not hurt at 4×4 — checked 4 times

a correlation the family is conditioned at

a grid the dense two-dimensional solve is affordable on

a two-dimensional kernel this file builds

and helps on at least one of the grids drawn

and the two-dimensional one is not

enough grids to show a trend

the one-dimensional count is flat

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.

Structure, and the solver that cannot see it

Two dimensions, and the cluster that thins

The same kernel, the same averaging, the same transform — applied along two axes instead of one. In one dimension the preconditioned step count is 7, 10, 10, 10; on square grids with the same unknown counts it is 10, 18, 20, 21, and the share of the spectrum near one falls from 56% to 17%.

The whole library · All essays · What must fail