two-residuals

At its defaults it draws the residual conjugate gradients reports and the residual of the vector it holds, along one run. A 50×50 diagonal matrix with one eigenvalue at 10⁻¹⁴ and the rest between 1 and 50, so every product with it is exact to a rounding and nothing below can be blamed on the operator. The lower curve is r ← r − αAp, updated by the recurrence, which is what a stopping test reads. The upper curve is ‖b − Ax‖/‖b‖, recomputed from the iterate. They start as the same vector and end 7.3·10¹⁰ apart, with the reported one at 6.92·10⁻²¹ — below the unit roundoff of 1.11·10⁻¹⁶, which is the shortest proof available that it is not the residual of anything.

two-residuals is one function in lib/figures/resgap.js — residual gap — the number a method reports against the residual of its 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.

A 50×50 diagonal matrix with one eigenvalue at 10⁻¹⁴ and the rest between 1 and 50, so every product with it is exact to a rounding and nothing below can be blamed on the operator. The lower curve is r ← r − αAp, updated by the recurrence, which is what a stopping test reads. The upper curve is ‖b − Ax‖/‖b‖, recomputed from the iterate. They start as the same vector and end 7.3·10¹⁰ apart, with the reported one at 6.92·10⁻²¹ — below the unit roundoff of 1.11·10⁻¹⁶, which is the shortest proof available that it is not the residual of anything.

A 50×50 diagonal matrix with one eigenvalue at 10⁻¹⁴ and the rest between 1 and 50, so every product with it is exact to a rounding and nothing below can be blamed on the operator. The lower curve is r ← r − αAp, updated by the recurrence, which is what a stopping test reads. The upper curve is ‖b − Ax‖/‖b‖, recomputed from the iterate. They start as the same vector and end 7.3·10¹⁰ apart, with the reported one at 6.92·10⁻²¹ — below the unit roundoff of 1.11·10⁻¹⁶, which is the shortest proof available that it is not the residual of anything.

A 50×50 diagonal matrix with one eigenvalue at 10⁻⁶ and the rest between 1 and 50, so every product with it is exact to a rounding and nothing below can be blamed on the operator. The lower curve is r ← r − αAp, updated by the recurrence, which is what a stopping test reads. The upper curve is ‖b − Ax‖/‖b‖, recomputed from the iterate. They start as the same vector and end 7.1·10⁶ apart, with the reported one at 7.26·10⁻²¹ — below the unit roundoff of 1.11·10⁻¹⁶, which is the shortest proof available that it is not the residual of anything.

A 50×50 diagonal matrix with one eigenvalue at 10⁻⁴ and the rest between 1 and 50, so every product with it is exact to a rounding and nothing below can be blamed on the operator. The lower curve is r ← r − αAp, updated by the recurrence, which is what a stopping test reads. The upper curve is ‖b − Ax‖/‖b‖, recomputed from the iterate. They start as the same vector and end 7.8·10⁵ apart, with the reported one at 5.89·10⁻²¹ — below the unit roundoff of 1.11·10⁻¹⁶, which is the shortest proof available that it is not the residual of anything.

A 50×50 diagonal matrix with one eigenvalue at 10⁻¹⁰ and the rest between 1 and 50, so every product with it is exact to a rounding and nothing below can be blamed on the operator. The lower curve is r ← r − αAp, updated by the recurrence, which is what a stopping test reads. The upper curve is ‖b − Ax‖/‖b‖, recomputed from the iterate. They start as the same vector and end 8.6·10⁸ apart, with the reported one at 6.89·10⁻²¹ — below the unit roundoff of 1.11·10⁻¹⁶, which is the shortest proof available that it is not the residual of anything.

A 50×50 diagonal matrix with one eigenvalue at 10⁻⁸ and the rest between 1 and 50, so every product with it is exact to a rounding and nothing below can be blamed on the operator. The lower curve is r ← r − αAp, updated by the recurrence, which is what a stopping test reads. The upper curve is ‖b − Ax‖/‖b‖, recomputed from the iterate. They start as the same vector and end 7.5·10⁷ apart, with the reported one at 7.91·10⁻²¹ — below the unit roundoff of 1.11·10⁻¹⁶, which is the shortest proof available that it is not the residual of anything.

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.

6 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 113 of 238 generators — 98 print a residual and 15 are exempt with a published reason; 125 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.