Backward error of Cramer's rule and of elimination on 11974 2×2 systems at 24 bits

At its defaults it draws backward error of cramer's rule and of elimination on 11974 2×2 systems at 24 bits. The share of systems whose normwise backward error exceeds a given multiple of the unit roundoff. Elimination never exceeds 1.2u. Cramer's rule reaches 395u on the same systems, on a matrix whose condition number is 3.39·10⁶ and which elimination solved to 0.3u. The control curve, over uniformly random 2×2 systems, stops at 4.9u — so the failure belongs to the near-parallel family and not to a choice of scale.

cramer-backward is one function in lib/figures/determinant.js — the determinant — the scalar that answers none of the questions it is asked. 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 share of systems whose normwise backward error exceeds a given multiple of the unit roundoff. Elimination never exceeds 1.2u. Cramer's rule reaches 395u on the same systems, on a matrix whose condition number is 3.39·10⁶ and which elimination solved to 0.3u. The control curve, over uniformly random 2×2 systems, stops at 4.9u — so the failure belongs to the near-parallel family and not to a choice of scale.

The share of systems whose normwise backward error exceeds a given multiple of the unit roundoff. Elimination never exceeds 1.2u. Cramer's rule reaches 395u on the same systems, on a matrix whose condition number is 3.39·10⁶ and which elimination solved to 0.3u. The control curve, over uniformly random 2×2 systems, stops at 4.9u — so the failure belongs to the near-parallel family and not to a choice of scale.

The share of systems whose normwise backward error exceeds a given multiple of the unit roundoff. Elimination never exceeds 1.1u. Cramer's rule reaches 63u on the same systems, on a matrix whose condition number is 6.98·10⁴ and which elimination solved to 0.0u. The control curve, over uniformly random 2×2 systems, stops at 3.0u — so the failure belongs to the near-parallel family and not to a choice of scale.

The share of systems whose normwise backward error exceeds a given multiple of the unit roundoff. Elimination never exceeds 1.2u. Cramer's rule reaches 229u on the same systems, on a matrix whose condition number is 3.37·10⁶ and which elimination solved to 0.0u. The control curve, over uniformly random 2×2 systems, stops at 3.9u — so the failure belongs to the near-parallel family and not to a choice of scale.

The share of systems whose normwise backward error exceeds a given multiple of the unit roundoff. Elimination never exceeds 1.2u. Cramer's rule reaches 214u on the same systems, on a matrix whose condition number is 3.37·10⁶ and which elimination solved to 0.2u. The control curve, over uniformly random 2×2 systems, stops at 4.6u — so the failure belongs to the near-parallel family and not to a choice of scale.

The share of systems whose normwise backward error exceeds a given multiple of the unit roundoff. Elimination never exceeds 1.4u. Cramer's rule reaches 248u on the same systems, on a matrix whose condition number is 3.37·10⁶ and which elimination solved to 0.3u. The control curve, over uniformly random 2×2 systems, stops at 6.9u — so the failure belongs to the near-parallel family and not to a choice of scale.

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.