mgw-spectrum

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.

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.

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 9 distinct values and the largest distance from the closed form anywhere is 2.955. A minimal polynomial of degree three means a Krylov method finishes in three steps, which is what the next figure measures.

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 = 8, and (1 ± √5)/2 with multiplicity 2 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 1.354·10⁻¹⁴. A minimal polynomial of degree three means a Krylov method finishes in three steps, which is what the next figure measures.

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 9 distinct values and the largest distance from the closed form anywhere is 2.64. A minimal polynomial of degree three means a Krylov method finishes in three steps, which is what the next figure measures.

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 6.617·10⁻¹⁴. A minimal polynomial of degree three means a Krylov method finishes in three steps, which is what the next figure measures.

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions 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.

13 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 165 of 306 generators — 150 print a residual and 15 are exempt with a published reason; 141 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.