series-cancellation

At its defaults it draws ‖aᵏ/k!‖ against ‖e^a‖ for moler and van loan's 2×2, scaled by 1. The terms of the Taylor series rise to 1.4·10⁷ at k = 16 and fall back; the sum they produce has norm 2.58, drawn as the flat line. The ratio is 5.42·10⁶, so that many digits are cancelled away before truncation is a question, and the computed exponential has a relative error of 5.2·10⁻⁹ after 77 terms. Scaling and squaring on the same matrix returns 8.5·10⁻¹⁴.

series-cancellation is one function in lib/figures/funm.js — matrix functions — the definition that is not a method, and the vector that was wanted. 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 terms of the Taylor series rise to 1.4·10⁷ at k = 16 and fall back; the sum they produce has norm 2.58, drawn as the flat line. The ratio is 5.42·10⁶, so that many digits are cancelled away before truncation is a question, and the computed exponential has a relative error of 5.2·10⁻⁹ after 77 terms. Scaling and squaring on the same matrix returns 8.5·10⁻¹⁴.

The terms of the Taylor series rise to 1.4·10⁷ at k = 16 and fall back; the sum they produce has norm 2.58, drawn as the flat line. The ratio is 5.42·10⁶, so that many digits are cancelled away before truncation is a question, and the computed exponential has a relative error of 5.2·10⁻⁹ after 77 terms. Scaling and squaring on the same matrix returns 8.5·10⁻¹⁴.

The terms of the Taylor series rise to 81.6 at k = 4 and fall back; the sum they produce has norm 5.37, drawn as the flat line. The ratio is 15.2, so that many digits are cancelled away before truncation is a question, and the computed exponential has a relative error of 1.87·10⁻¹⁵ after 36 terms. Scaling and squaring on the same matrix returns 1.65·10⁻¹⁴.

The terms of the Taylor series rise to 2.39·10¹⁴ at k = 33 and fall back; the sum they produce has norm 1.01, drawn as the flat line. The ratio is 2.36·10¹⁴, so that many digits are cancelled away before truncation is a question, and the computed exponential has a relative error of 0.107 after 127 terms. Scaling and squaring on the same matrix returns 1.72·10⁻¹³.

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.

7 distinct claims across 4 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 92 of 198 generators — 77 print a residual and 15 are exempt with a published reason; 106 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.