The accuracy worth paying for in a preconditioner, against the condition number of the problem

At its defaults it draws the accuracy worth paying for in a preconditioner, against the condition number of the problem. Four complete cost sweeps, each reduced to the ε that minimised the total work, against the κ of the problem it was run on. The shift that moves κ over 2.7 decades leaves every distance between every pair of clusters exactly where it was — and the ranks the partition asks for at a fixed tolerance do not stay put: they fall 10, 9, 7, 5 across the sweep, which is about half. So the shift is not the purely spectral knob it was chosen to be; adding it to the diagonal makes the kernel decay faster as well, and the second effect is doing part of the work the first is credited with. The optimum walks 0.5 → 0.5 → 10⁻⁶ → 10⁻⁸ — six decades of accuracy, bought because the problem got harder and for no other reason. On the easiest problem the best hierarchical preconditioner in the comparison has rank one; on the hardest it is the tightest one on the sweep. How accurate an approximate inverse should be is a question with an answer, and the answer is not in the matrix's structure.

accuracy-optimum is one function in lib/figures/hsolve.js — solving with it — the recursion that assembles nothing, and the backward error that was chosen. 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.

Four complete cost sweeps, each reduced to the ε that minimised the total work, against the κ of the problem it was run on. The shift that moves κ over 2.7 decades leaves every distance between every pair of clusters exactly where it was — and the ranks the partition asks for at a fixed tolerance do not stay put: they fall 10, 9, 7, 5 across the sweep, which is about half. So the shift is not the purely spectral knob it was chosen to be; adding it to the diagonal makes the kernel decay faster as well, and the second effect is doing part of the work the first is credited with. The optimum walks 0.5 → 0.5 → 10⁻⁶ → 10⁻⁸ — six decades of accuracy, bought because the problem got harder and for no other reason. On the easiest problem the best hierarchical preconditioner in the comparison has rank one; on the hardest it is the tightest one on the sweep. How accurate an approximate inverse should be is a question with an answer, and the answer is not in the matrix's structure.

Four complete cost sweeps, each reduced to the ε that minimised the total work, against the κ of the problem it was run on. The shift that moves κ over 2.7 decades leaves every distance between every pair of clusters exactly where it was — and the ranks the partition asks for at a fixed tolerance do not stay put: they fall 10, 9, 7, 5 across the sweep, which is about half. So the shift is not the purely spectral knob it was chosen to be; adding it to the diagonal makes the kernel decay faster as well, and the second effect is doing part of the work the first is credited with. The optimum walks 0.5 → 0.5 → 10⁻⁶ → 10⁻⁸ — six decades of accuracy, bought because the problem got harder and for no other reason. On the easiest problem the best hierarchical preconditioner in the comparison has rank one; on the hardest it is the tightest one on the sweep. How accurate an approximate inverse should be is a question with an answer, and the answer is not in the matrix's structure.

Four complete cost sweeps, each reduced to the ε that minimised the total work, against the κ of the problem it was run on. The shift that moves κ over 2.6 decades leaves every distance between every pair of clusters exactly where it was — and the ranks the partition asks for at a fixed tolerance do not stay put: they fall 9, 8, 6, 4 across the sweep, which is about half. So the shift is not the purely spectral knob it was chosen to be; adding it to the diagonal makes the kernel decay faster as well, and the second effect is doing part of the work the first is credited with. The optimum walks 0.5 → 0.5 → 0.5 → 10⁻⁴ — six decades of accuracy, bought because the problem got harder and for no other reason. On the easiest problem the best hierarchical preconditioner in the comparison has rank one; on the hardest it is the tightest one on the sweep. How accurate an approximate inverse should be is a question with an answer, and the answer is not in the matrix's structure.

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.

12 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.

It draws a decomposition and prints its residual. It calls conditionSweep, conditionSweep, 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.