relative-spectrum

At its defaults it draws relative error of each singular value of a bidiagonal matrix graded over 30 decades, by four routes. The matrix is 8×8, bidiagonal, and every entry is a power of two times a small integer, so the matrix the float routines are handed and the matrix the exact routine reasons about are the same matrix. The reference is a Sturm bisection in BigInt rationals, which never rounds. σ_min is 2.098·10⁻³⁰. One-sided Jacobi holds every value to 4.41·10⁻¹⁶ relative and the zero-shift bidiagonal sweep to 2.21·10⁻¹⁶; the eigenvalues of BᵀB are wrong by 1 — not a rounding error but the whole number — and the damage is entirely at the small end.

relative-spectrum is one function in lib/figures/relacc.js — relative accuracy — small compared to what, against an exact rational answer. 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 matrix is 8×8, bidiagonal, and every entry is a power of two times a small integer, so the matrix the float routines are handed and the matrix the exact routine reasons about are the same matrix. The reference is a Sturm bisection in BigInt rationals, which never rounds. σ_min is 2.098·10⁻³⁰. One-sided Jacobi holds every value to 4.41·10⁻¹⁶ relative and the zero-shift bidiagonal sweep to 2.21·10⁻¹⁶; the eigenvalues of BᵀB are wrong by 1 — not a rounding error but the whole number — and the damage is entirely at the small end.

The matrix is 8×8, bidiagonal, and every entry is a power of two times a small integer, so the matrix the float routines are handed and the matrix the exact routine reasons about are the same matrix. The reference is a Sturm bisection in BigInt rationals, which never rounds. σ_min is 2.098·10⁻³⁰. One-sided Jacobi holds every value to 4.41·10⁻¹⁶ relative and the zero-shift bidiagonal sweep to 2.21·10⁻¹⁶; the eigenvalues of BᵀB are wrong by 1 — not a rounding error but the whole number — and the damage is entirely at the small end.

The matrix is 8×8, bidiagonal, and every entry is a power of two times a small integer, so the matrix the float routines are handed and the matrix the exact routine reasons about are the same matrix. The reference is a Sturm bisection in BigInt rationals, which never rounds. σ_min is 0.00478. One-sided Jacobi holds every value to 1.2·10⁻¹⁵ relative and the zero-shift bidiagonal sweep to 1.09·10⁻¹⁵; the eigenvalues of BᵀB are wrong by 3.63·10⁻¹⁴ — not a rounding error but the whole number — and the damage is entirely at the small end.

The matrix is 8×8, bidiagonal, and every entry is a power of two times a small integer, so the matrix the float routines are handed and the matrix the exact routine reasons about are the same matrix. The reference is a Sturm bisection in BigInt rationals, which never rounds. σ_min is 3.726·10⁻⁴⁵. One-sided Jacobi holds every value to 2.2·10⁻¹⁶ relative and the zero-shift bidiagonal sweep to 2.19·10⁻¹⁶; the eigenvalues of BᵀB are wrong by 2.37·10⁶ — not a rounding error but the whole number — and the damage is entirely at the small end.

The matrix is 8×8, bidiagonal, and every entry is a power of two times a small integer, so the matrix the float routines are handed and the matrix the exact routine reasons about are the same matrix. The reference is a Sturm bisection in BigInt rationals, which never rounds. σ_min is 1.934·10⁻¹¹. One-sided Jacobi holds every value to 7.1·10⁻¹⁶ relative and the zero-shift bidiagonal sweep to 4.41·10⁻¹⁶; the eigenvalues of BᵀB are wrong by 14.5 — not a rounding error but the whole number — and the damage is entirely at the small end.

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.

11 distinct claims across 5 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 routeComparison, 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 113 of 219 generators — 98 print a residual and 15 are exempt with a published reason; 106 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.