What a hierarchical preconditioner costs, against how accurate it is, at κ = 2.09e+1

At its defaults it draws what a hierarchical preconditioner costs, against how accurate it is, at κ = 2.09e+1. The iteration is conjugate gradients on the true matrix and the preconditioner is a solve with the representation, so what is measured is how much of the compression the outer loop repairs and what the repair costs. The count falls monotonically — 13, 8, 6, 3, 2, 2 steps against 41 with no preconditioner — and it keeps falling all the way to the tightest accuracy, which is the number a solver prints and the one everybody watches. The total work, one dense product with A plus one solve with the compressed matrix per step, is the other curve, and its minimum is at ε = 0.5 where the preconditioner has rank 1. Past that point every extra digit in the preconditioner is bought and not needed: at the tightest setting the run costs 2.42 times the cheapest, having taken 11 fewer steps to get there.

precondition-total 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.

The iteration is conjugate gradients on the true matrix and the preconditioner is a solve with the representation, so what is measured is how much of the compression the outer loop repairs and what the repair costs. The count falls monotonically — 13, 8, 6, 3, 2, 2 steps against 41 with no preconditioner — and it keeps falling all the way to the tightest accuracy, which is the number a solver prints and the one everybody watches. The total work, one dense product with A plus one solve with the compressed matrix per step, is the other curve, and its minimum is at ε = 0.5 where the preconditioner has rank 1. Past that point every extra digit in the preconditioner is bought and not needed: at the tightest setting the run costs 2.42 times the cheapest, having taken 11 fewer steps to get there.

The iteration is conjugate gradients on the true matrix and the preconditioner is a solve with the representation, so what is measured is how much of the compression the outer loop repairs and what the repair costs. The count falls monotonically — 13, 8, 6, 3, 2, 2 steps against 41 with no preconditioner — and it keeps falling all the way to the tightest accuracy, which is the number a solver prints and the one everybody watches. The total work, one dense product with A plus one solve with the compressed matrix per step, is the other curve, and its minimum is at ε = 0.5 where the preconditioner has rank 1. Past that point every extra digit in the preconditioner is bought and not needed: at the tightest setting the run costs 2.42 times the cheapest, having taken 11 fewer steps to get there.

The iteration is conjugate gradients on the true matrix and the preconditioner is a solve with the representation, so what is measured is how much of the compression the outer loop repairs and what the repair costs. The count falls monotonically — 17, 17, 8, 4, 2, 2 steps against 105 with no preconditioner — and it keeps falling all the way to the tightest accuracy, which is the number a solver prints and the one everybody watches. The total work, one dense product with A plus one solve with the compressed matrix per step, is the other curve, and its minimum is at ε = 10⁻⁶ where the preconditioner has rank 6. Past that point every extra digit in the preconditioner is bought and not needed: at the tightest setting the run costs 1.34 times the cheapest, having taken 0 fewer steps to get there.

The iteration is conjugate gradients on the true matrix and the preconditioner is a solve with the representation, so what is measured is how much of the compression the outer loop repairs and what the repair costs. The count falls monotonically — 16, 14, 7, 4, 3, 2 steps against 72 with no preconditioner — and it keeps falling all the way to the tightest accuracy, which is the number a solver prints and the one everybody watches. The total work, one dense product with A plus one solve with the compressed matrix per step, is the other curve, and its minimum is at ε = 0.5 where the preconditioner has rank 1. Past that point every extra digit in the preconditioner is bought and not needed: at the tightest setting the run costs 1.59 times the cheapest, having taken 14 fewer steps to get there.

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.

3 distinct claims across 4 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 preconditionSweep, 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.