The forward error of a hierarchical solve, and κ times the backward error that was chosen for it

At its defaults it draws the forward error of a hierarchical solve, and κ times the backward error that was chosen for it. Forward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 24.39 is a property of the problem — the same at every leaf size, because a partition is not a conditioning — and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.981 and sits 14× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.

forward-predicted is one function in lib/figures/hsolve.js — solving with it — the recursion that assembles nothing, and the backward error that was chosen. 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.

Forward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 24.39 is a property of the problem — the same at every leaf size, because a partition is not a conditioning — and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.981 and sits 14× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.

Forward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 20.88 is a property of the problem — the same at every leaf size, because a partition is not a conditioning — and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.971 and sits 16× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.

Forward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 17.43 is a property of the problem — the same at every leaf size, because a partition is not a conditioning — and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.970 and sits 12× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.

Forward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 24.39 is a property of the problem — the same at every leaf size, because a partition is not a conditioning — and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.981 and sits 14× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.

Forward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 20.88 is a property of the problem — the same at every leaf size, because a partition is not a conditioning — and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.997 and sits 10× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.

Forward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 20.88 is a property of the problem — the same at every leaf size, because a partition is not a conditioning — and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.972 and sits 18× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.

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 accuracySweep, 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.