The growth factor of a 12×12 elimination, against the condition number of the matrix

At its defaults it draws the growth factor of a 12×12 elimination, against the condition number of the matrix. Three curves against κ. Cholesky's growth factor is exactly 1 at every condition number drawn — the elimination never produces an entry larger than the matrix already had. Partial pivoting on the same matrices reaches the same growth, and reaches it by making up to 9 row interchanges, each of which destroys the symmetry that was the reason to use a symmetric factorisation. The bound the general theory allows, 2^11 = 2048, is drawn above them both.

cholesky-growth is one function in lib/figures/symmetric.js — symmetric eliminations — a growth factor of exactly one, and a diagonal that is zero. 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.

Three curves against κ. Cholesky's growth factor is exactly 1 at every condition number drawn — the elimination never produces an entry larger than the matrix already had. Partial pivoting on the same matrices reaches the same growth, and reaches it by making up to 9 row interchanges, each of which destroys the symmetry that was the reason to use a symmetric factorisation. The bound the general theory allows, 2^11 = 2048, is drawn above them both.

On the planted family, the magnitude of the Rayleigh quotient over the smallest eigenvalue, −8.40 at every coupling, for the direction each rule's factors give, and the magnitude of the eigenvalue nearest zero over the same, on logarithmic axes. Bunch–Kaufman's direction falls as the square of the coupling; the bounded rule's holds 0.14 at every coupling; and the eigenvalue nearest zero — the one inverse iteration with either factorisation lands on within two steps — falls as the square of the coupling and is positive until it reaches the rounding level.

The planted 24 × 24 matrix at coupling 10 to the minus 8. From the direction each rule's factors give — Bunch–Kaufman's holding 6.9e-16 of the curvature, the bounded rule's 0.14 — three refinements: inverse iteration with the same factors, power iteration on the norm times the identity less the matrix, and Lanczos. Lanczos reaches ninety-nine per cent in 9 products from Bunch–Kaufman's direction and 6 from the bounded rule's; power iteration reaches ninety per cent in 13 from the bounded rule's and not within sixty from Bunch–Kaufman's; inverse iteration leaves both at the eigenvalue nearest zero, which is positive. Quotients at or below zero are drawn on the floor.

On the planted family. Lanczos needs 7 products from Bunch–Kaufman's direction and 4 from the bounded rule's at every coupling. Power iteration from the bounded rule's needs 13 at every coupling; from Bunch–Kaufman's it needs 28, 39, 54, more than sixty, more than sixty as the coupling falls through the five values. Points drawn at the top did not arrive within sixty products.

Random 24 × 24 symmetric matrices with standard normal entries, factored by Bunch–Kaufman and by the bounded rule. The Rayleigh quotient over the smallest eigenvalue: the factors' own direction has a median of 0.077 and 0.103; after eight steps of inverse iteration with the same factors, 0.000 and 0.006; after eight Lanczos products, 0.996 and 0.996, the least of all sixty 0.749.

The planted matrix at coupling ten to the minus eight, smallest eigenvalue −8.399. The shift is the most negative eigenvalue of D: −7.339 for Bunch–Kaufman's factors and −5.437 for the bounded rule's, then each pushed a tenth further. Bunch–Kaufman, shift −7.34: the quotient settles at 0.849 of the smallest eigenvalue, the eigenvalue nearest the shift being −7.135; Bunch–Kaufman, shift −8.07: the quotient settles at 1.000 of the smallest eigenvalue, the eigenvalue nearest the shift being −8.399; bounded rule, shift −5.44: the quotient settles at 0.597 of the smallest eigenvalue, the eigenvalue nearest the shift being −5.015; bounded rule, shift −5.98: the quotient settles at 0.598 of the smallest eigenvalue, the eigenvalue nearest the shift being −5.015.

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.

37 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 luFactor, choleskyGrowth, 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.