pencil-routes

At its defaults it draws relative eigenvalue error of two reductions of ax = λbx, against κ(b), n = 12. The pencil is built as B = LLᵀ and A = LΛLᵀ, so its generalised eigenvalues are exactly Λ and the error is a measurement rather than a comparison. Forming B⁻¹A gives a matrix whose departure from symmetry reaches 1.08 — about its own size — and reducing through the Cholesky factor gives one that is nearer symmetric by a factor of at least 6078. That difference is real and it does not appear in the accuracy: the two error curves have fitted slopes of 0.92 and 0.98 against κ(B) and stay within a factor of 2.3 of each other over fifteen decades. The conditioning belongs to the pencil, and no reduction of it escapes.

pencil-routes is one function in lib/figures/pencil.js — pencils — two matrices, an eigenvalue with no value, and a problem with no 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.

The pencil is built as B = LLᵀ and A = LΛLᵀ, so its generalised eigenvalues are exactly Λ and the error is a measurement rather than a comparison. Forming B⁻¹A gives a matrix whose departure from symmetry reaches 1.08 — about its own size — and reducing through the Cholesky factor gives one that is nearer symmetric by a factor of at least 6078. That difference is real and it does not appear in the accuracy: the two error curves have fitted slopes of 0.92 and 0.98 against κ(B) and stay within a factor of 2.3 of each other over fifteen decades. The conditioning belongs to the pencil, and no reduction of it escapes.

The pencil is built as B = LLᵀ and A = LΛLᵀ, so its generalised eigenvalues are exactly Λ and the error is a measurement rather than a comparison. Forming B⁻¹A gives a matrix whose departure from symmetry reaches 1.08 — about its own size — and reducing through the Cholesky factor gives one that is nearer symmetric by a factor of at least 6078. That difference is real and it does not appear in the accuracy: the two error curves have fitted slopes of 0.92 and 0.98 against κ(B) and stay within a factor of 2.3 of each other over fifteen decades. The conditioning belongs to the pencil, and no reduction of it escapes.

The pencil is built as B = LLᵀ and A = LΛLᵀ, so its generalised eigenvalues are exactly Λ and the error is a measurement rather than a comparison. Forming B⁻¹A gives a matrix whose departure from symmetry reaches 1.07 — about its own size — and reducing through the Cholesky factor gives one that is nearer symmetric by a factor of at least 2.7·10⁶. That difference is real and it does not appear in the accuracy: the two error curves have fitted slopes of 1.04 and 1.01 against κ(B) and stay within a factor of 4.9 of each other over fifteen decades. The conditioning belongs to the pencil, and no reduction of it escapes.

The pencil is built as B = LLᵀ and A = LΛLᵀ, so its generalised eigenvalues are exactly Λ and the error is a measurement rather than a comparison. Forming B⁻¹A gives a matrix whose departure from symmetry reaches 1.29 — about its own size — and reducing through the Cholesky factor gives one that is nearer symmetric by a factor of at least 643. That difference is real and it does not appear in the accuracy: the two error curves have fitted slopes of 0.92 and 0.96 against κ(B) and stay within a factor of 19.9 of each other over fifteen decades. The conditioning belongs to the pencil, and no reduction of it escapes.

The pencil is built as B = LLᵀ and A = LΛLᵀ, so its generalised eigenvalues are exactly Λ and the error is a measurement rather than a comparison. Forming B⁻¹A gives a matrix whose departure from symmetry reaches 1.33 — about its own size — and reducing through the Cholesky factor gives one that is nearer symmetric by a factor of at least 2.1·10⁶. That difference is real and it does not appear in the accuracy: the two error curves have fitted slopes of 0.89 and 0.90 against κ(B) and stay within a factor of 5.0 of each other over fifteen decades. The conditioning belongs to the pencil, and no reduction of it escapes.

The pencil is built as B = LLᵀ and A = LΛLᵀ, so its generalised eigenvalues are exactly Λ and the error is a measurement rather than a comparison. Forming B⁻¹A gives a matrix whose departure from symmetry reaches 1.29 — about its own size — and reducing through the Cholesky factor gives one that is nearer symmetric by a factor of at least 1331. That difference is real and it does not appear in the accuracy: the two error curves have fitted slopes of 0.93 and 0.99 against κ(B) and stay within a factor of 54.3 of each other over fifteen decades. The conditioning belongs to the pencil, and no reduction of it escapes.

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.

19 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 conditioningSweep, 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 113 of 219 generators — 98 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.