Incomplete Cholesky on the 10×10 grid: κ 48.4 → 5.12
At its defaults it draws incomplete cholesky on the 10×10 grid: κ 48.4 → 5.12. A semi-logarithmic plot of relative residual against iteration for conjugate gradients with and without an incomplete Cholesky preconditioner, the preconditioned curve falling faster.
preconditioner-effect is one function in lib/figures/iterative.js —
iterative — krylov and stationary methods against rates known in closed form. 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 semi-logarithmic plot of relative residual against iteration for conjugate gradients with and without an incomplete Cholesky preconditioner, the preconditioned curve falling faster.
grid: 10
The arguments are the ones A rate that is known in advance passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A semi-logarithmic plot of relative residual against iteration for conjugate gradients with and without an incomplete Cholesky preconditioner, the preconditioned curve falling faster.
grid: 6
The arguments are the ones Changing the condition number on purpose passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A semi-logarithmic plot of relative residual against iteration for conjugate gradients with and without an incomplete Cholesky preconditioner, the preconditioned curve falling faster.
grid: 14
The arguments are the ones Changing the condition number on purpose passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A semi-logarithmic plot of relative residual against iteration for conjugate gradients with and without an incomplete Cholesky preconditioner, the preconditioned curve falling faster.
grid: 9
The arguments are the ones Changing the condition number on purpose passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A semi-logarithmic plot of relative residual against iteration for conjugate gradients with and without an incomplete Cholesky preconditioner, the preconditioned curve falling faster.
grid: 12
The arguments are the ones Changing the condition number on purpose passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A semi-logarithmic plot of relative residual against iteration for conjugate gradients with and without an incomplete Cholesky preconditioner, the preconditioned curve falling faster.
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.
5 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 the iteration count falls with it
matmul shapes agree
the factorisation really is incomplete
the incomplete factorisation exists
the preconditioned condition number is smaller
Against the rule
It draws a decomposition and prints its residual. It calls
ichol,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
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.
A rate that is known in advance
On the model problem, Jacobi contracts by cos(π/(n+1)) per step, Gauss–Seidel by its square, and optimally relaxed SOR by a number given in closed form. Three rates, all known before anything runs, and all measurable against what runs.
Iterating, instead of factorisingChanging the condition number on purpose
Preconditioning is usually introduced as a trick that makes an iteration converge faster. It is not a trick. It is solving a different system with the same solution and a condition number chosen rather than inherited, and the new condition number is computable.