An exact solve of a measured 8 × 8 system, against the error the measurement put there

At its defaults it draws an exact solve of a measured 8 × 8 system, against the error the measurement put there. The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-6 the answer is wrong by 1.342, which is 1.3·10⁶ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

inexact-data is one function in lib/figures/integer.js — exact arithmetic — no residual to print, and a cost measured in the length of the numbers. 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 rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-6 the answer is wrong by 1.342, which is 1.3·10⁶ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-12 the answer is wrong by 9.988·10⁻⁴, which is 10·10⁸ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-6 the answer is wrong by 2.049·10⁻⁶, which is 2.05 times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-14 the answer is wrong by 1.002·10⁻⁵, which is 10⁹ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-10 the answer is wrong by 0.07441, which is 7.4·10⁸ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-8 the answer is wrong by 0.2705, which is 2.7·10⁷ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

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.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

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.