The two perturbations that make one computed solution exact, on a 10×10 Toeplitz system

At its defaults it draws the two perturbations that make one computed solution exact, on a 10×10 toeplitz system. A Kac–Murdock–Szegő matrix at ρ = 0.95, solved by Levinson's recursion. Every backward-error claim on this site says the computed answer solves a nearby problem exactly, and the nearby problem is the top matrix: the smallest perturbation of any kind, 1.89·10⁻¹⁷ relative, rank one, and constant along 0.97 of the way to none of its diagonals. The bottom matrix is the smallest perturbation that is itself a symmetric Toeplitz matrix — the same kind of object the problem was posed with — and it is 2.65·10⁻¹³, larger by a factor of 1.41·10⁴. Both explain the same computed answer exactly. Only one of them is a problem anybody could have posed.

perturbation-shape is one function in lib/figures/structbe.js — structured backward error — whether the nearby problem is the same kind of problem. 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.

A Kac–Murdock–Szegő matrix at ρ = 0.95, solved by Levinson's recursion. Every backward-error claim on this site says the computed answer solves a nearby problem exactly, and the nearby problem is the top matrix: the smallest perturbation of any kind, 1.89·10⁻¹⁷ relative, rank one, and constant along 0.97 of the way to none of its diagonals. The bottom matrix is the smallest perturbation that is itself a symmetric Toeplitz matrix — the same kind of object the problem was posed with — and it is 2.65·10⁻¹³, larger by a factor of 1.41·10⁴. Both explain the same computed answer exactly. Only one of them is a problem anybody could have posed.

A Kac–Murdock–Szegő matrix at ρ = 0.95, solved by Levinson's recursion. Every backward-error claim on this site says the computed answer solves a nearby problem exactly, and the nearby problem is the top matrix: the smallest perturbation of any kind, 1.89·10⁻¹⁷ relative, rank one, and constant along 0.97 of the way to none of its diagonals. The bottom matrix is the smallest perturbation that is itself a symmetric Toeplitz matrix — the same kind of object the problem was posed with — and it is 2.65·10⁻¹³, larger by a factor of 1.41·10⁴. Both explain the same computed answer exactly. Only one of them is a problem anybody could have posed.

A Kac–Murdock–Szegő matrix at ρ = 0.5, solved by Levinson's recursion. Every backward-error claim on this site says the computed answer solves a nearby problem exactly, and the nearby problem is the top matrix: the smallest perturbation of any kind, 2.17·10⁻¹⁷ relative, rank one, and constant along 0.93 of the way to none of its diagonals. The bottom matrix is the smallest perturbation that is itself a symmetric Toeplitz matrix — the same kind of object the problem was posed with — and it is 3.83·10⁻¹⁵, larger by a factor of 177. Both explain the same computed answer exactly. Only one of them is a problem anybody could have posed.

A Kac–Murdock–Szegő matrix at ρ = 0.7, solved by Levinson's recursion. Every backward-error claim on this site says the computed answer solves a nearby problem exactly, and the nearby problem is the top matrix: the smallest perturbation of any kind, 1.67·10⁻¹⁷ relative, rank one, and constant along 0.88 of the way to none of its diagonals. The bottom matrix is the smallest perturbation that is itself a symmetric Toeplitz matrix — the same kind of object the problem was posed with — and it is 6.83·10⁻¹⁵, larger by a factor of 409. Both explain the same computed answer exactly. Only one of them is a problem anybody could have posed.

A Kac–Murdock–Szegő matrix at ρ = 0.8, solved by Levinson's recursion. Every backward-error claim on this site says the computed answer solves a nearby problem exactly, and the nearby problem is the top matrix: the smallest perturbation of any kind, 1.99·10⁻¹⁷ relative, rank one, and constant along 0.95 of the way to none of its diagonals. The bottom matrix is the smallest perturbation that is itself a symmetric Toeplitz matrix — the same kind of object the problem was posed with — and it is 6.44·10⁻¹⁵, larger by a factor of 324. Both explain the same computed answer exactly. Only one of them is a problem anybody could have posed.

A Kac–Murdock–Szegő matrix at ρ = 0.9, solved by Levinson's recursion. Every backward-error claim on this site says the computed answer solves a nearby problem exactly, and the nearby problem is the top matrix: the smallest perturbation of any kind, 1.72·10⁻¹⁷ relative, rank one, and constant along 0.88 of the way to none of its diagonals. The bottom matrix is the smallest perturbation that is itself a symmetric Toeplitz matrix — the same kind of object the problem was posed with — and it is 1.01·10⁻¹³, larger by a factor of 5834. Both explain the same computed answer exactly. Only one of them is a problem anybody could have posed.

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.

7 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 levinson, structuredBackwardError, structuredBackwardError, 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.