The 16 eigenvalues of an 8 × 8 quadratic eigenvalue problem, computed and in closed form

At its defaults it draws the 16 eigenvalues of an 8 × 8 quadratic eigenvalue problem, computed and in closed form. λ²M + λC + K for a chain of 8 masses with C = 0.3M + 0.1K. The crosses are the closed form — one scalar quadratic per eigenvalue of K, whose roots are known exactly — and the discs are the eigenvalues a real Schur factorisation returns from the 16 × 16 first companion linearisation. There are 16 of them for a matrix with 8 rows, of which 16 are complex and arrive in conjugate pairs, so the eigenvectors cannot be independent: 9 vectors in 8 dimensions never are. The worst disagreement between the two routes is 2.12·10⁻¹⁵, and the routes share nothing but the three coefficient matrices.

qep-spectrum is one function in lib/figures/quadratic.js — a matrix that depends on its own eigenvalue — 2n answers for n rows, and a count that is a degree. 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.

λ²M + λC + K for a chain of 8 masses with C = 0.3M + 0.1K. The crosses are the closed form — one scalar quadratic per eigenvalue of K, whose roots are known exactly — and the discs are the eigenvalues a real Schur factorisation returns from the 16 × 16 first companion linearisation. There are 16 of them for a matrix with 8 rows, of which 16 are complex and arrive in conjugate pairs, so the eigenvectors cannot be independent: 9 vectors in 8 dimensions never are. The worst disagreement between the two routes is 2.12·10⁻¹⁵, and the routes share nothing but the three coefficient matrices.

λ²M + λC + K for a chain of 8 masses with C = 0.3M + 0.1K. The crosses are the closed form — one scalar quadratic per eigenvalue of K, whose roots are known exactly — and the discs are the eigenvalues a real Schur factorisation returns from the 16 × 16 first companion linearisation. There are 16 of them for a matrix with 8 rows, of which 16 are complex and arrive in conjugate pairs, so the eigenvectors cannot be independent: 9 vectors in 8 dimensions never are. The worst disagreement between the two routes is 2.12·10⁻¹⁵, and the routes share nothing but the three coefficient matrices.

λ²M + λC + K for a chain of 8 masses with C = 0.3M + 0K. The crosses are the closed form — one scalar quadratic per eigenvalue of K, whose roots are known exactly — and the discs are the eigenvalues a real Schur factorisation returns from the 16 × 16 first companion linearisation. There are 16 of them for a matrix with 8 rows, of which 16 are complex and arrive in conjugate pairs, so the eigenvectors cannot be independent: 9 vectors in 8 dimensions never are. The worst disagreement between the two routes is 4.14·10⁻¹⁵, and the routes share nothing but the three coefficient matrices.

λ²M + λC + K for a chain of 8 masses with C = 0.3M + 0.05K. The crosses are the closed form — one scalar quadratic per eigenvalue of K, whose roots are known exactly — and the discs are the eigenvalues a real Schur factorisation returns from the 16 × 16 first companion linearisation. There are 16 of them for a matrix with 8 rows, of which 16 are complex and arrive in conjugate pairs, so the eigenvectors cannot be independent: 9 vectors in 8 dimensions never are. The worst disagreement between the two routes is 1.72·10⁻¹⁵, and the routes share nothing but the three coefficient matrices.

λ²M + λC + K for a chain of 8 masses with C = 0.3M + 0.3K. The crosses are the closed form — one scalar quadratic per eigenvalue of K, whose roots are known exactly — and the discs are the eigenvalues a real Schur factorisation returns from the 16 × 16 first companion linearisation. There are 16 of them for a matrix with 8 rows, of which 16 are complex and arrive in conjugate pairs, so the eigenvectors cannot be independent: 9 vectors in 8 dimensions never are. The worst disagreement between the two routes is 3.05·10⁻¹⁵, and the routes share nothing but the three coefficient matrices.

λ²M + λC + K for a chain of 8 masses with C = 0.3M + 1K. The crosses are the closed form — one scalar quadratic per eigenvalue of K, whose roots are known exactly — and the discs are the eigenvalues a real Schur factorisation returns from the 16 × 16 first companion linearisation. There are 16 of them for a matrix with 8 rows, of which 12 are complex and arrive in conjugate pairs, so the eigenvectors cannot be independent: 9 vectors in 8 dimensions never are. The worst disagreement between the two routes is 9.24·10⁻¹⁵, and the routes share nothing but the three coefficient matrices.

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.

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 qepEigen, 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.