The spectrum of P⁻¹K with S = AH⁻¹Aᵀ, exactly, at 10 unknowns and 4 constraints

At its defaults it draws the spectrum of p⁻¹k with s = ah⁻¹aᵀ, exactly, at 10 unknowns and 4 constraints. P = blkdiag(H, S). With S the exact Schur complement AH⁻¹Aᵀ the preconditioned matrix has exactly three distinct eigenvalues — 1 with multiplicity n − m = 6, and (1 ± √5)/2 with multiplicity 4 each. Those are 1 − φ = -0.618034 and φ = 1.61803, the golden ratio, which arrives from λ² − λ − 1 = 0 rather than from anything anybody chose. The dashed lines are that closed form and the marks are the computed spectrum; here they are 3 distinct values and the largest distance from the closed form anywhere is 2.909·10⁻¹⁴. A minimal polynomial of degree three means a Krylov method finishes in three steps, which is what the next figure measures.

mgw-spectrum is one function in lib/figures/blockprec.js — preconditioning a saddle point — three eigenvalues, and a spectrum with no constraint in it. 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.

P = blkdiag(H, S). With S the exact Schur complement AH⁻¹Aᵀ the preconditioned matrix has exactly three distinct eigenvalues — 1 with multiplicity n − m = 6, and (1 ± √5)/2 with multiplicity 4 each. Those are 1 − φ = -0.618034 and φ = 1.61803, the golden ratio, which arrives from λ² − λ − 1 = 0 rather than from anything anybody chose. The dashed lines are that closed form and the marks are the computed spectrum; here they are 3 distinct values and the largest distance from the closed form anywhere is 2.909·10⁻¹⁴. A minimal polynomial of degree three means a Krylov method finishes in three steps, which is what the next figure measures.

κ(H) = 100, κ(A) = 10, exact Schur complement. The ten eigenvalues that were Jordan pairs sit about 0.52times the square root of τ from one; the other seven about 0.35τ; the residual after two steps is 7.4·10⁻¹¹ at τ = 10⁻¹⁰, 7.4·10⁻¹⁰ at τ = 10⁻⁹, 7.4·10⁻⁹ at τ = 10⁻⁸, 7.4·10⁻⁸ at τ = 10⁻⁷, 7.4·10⁻⁷ at τ = 10⁻⁶, 7.4·10⁻⁶ at τ = 10⁻⁵, 7.4·10⁻⁵ at τ = 10⁻⁴, 7.4·10⁻⁴ at τ = 0.001, 0.0073 at τ = 0.01. Eigenvalues are drawn from τ = 10⁻⁸ up: below it the Francis iteration does not converge on this matrix, and below 10⁻¹⁰ the residual sits at the rounding floor.

In units of the square root of τ the ten eigenvalues from the Jordan pairs sit between 0.1 and 0.7 from the centre at every τ the dial offers, and the other seven at 6.1e-3 and closer: they move like τ, so on this scale they fall into the centre as τ shrinks.

The split over the square root of τ, drawn from τ = 10⁻⁸ where the eigenvalues can be computed on every system, lies between 0.42 and 0.62 and is flat on every system. The two-step residual over τ, from 10⁻¹⁰, lies between 2.9·10⁻⁷ and 2.1; it is flat on the nine systems with κ(A) = 10 and rises with τ on the nine with κ(A) = 10⁴, where a term in τ squared overtakes a small linear one. Each system is a line.

τ = 10⁻¹²: GMRES 1, 0.25, 7.4·10⁻¹³ — 2 steps; MINRES 3 steps. τ = 10⁻⁸: GMRES 1, 0.25, 7.4·10⁻⁹, 4.1·10⁻¹⁰, 1.9·10⁻¹⁶ — 4 steps; MINRES 6 steps. τ = 10⁻⁴: GMRES 1, 0.25, 7.4·10⁻⁵, 4.1·10⁻⁶, 7.7·10⁻¹⁰, 8.9·10⁻¹¹ — 5 steps; MINRES 9 steps. τ = 0.01: GMRES 1, 0.25, 0.0073, 4·10⁻⁴, 7.2·10⁻⁶, 8.9·10⁻⁷, 1.9·10⁻⁸, 3.4·10⁻¹⁰, 9.7·10⁻¹² — 8 steps; MINRES 12 steps. The dotted line is the tolerance, 10⁻¹⁰.

234 cells; the rule is exact on 179 and within one step on 228. A circle's area is the number of cells at that pair.

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.

34 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.

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