null-basis-bars

At its defaults it draws three bases for the same null space, at 10 unknowns and 4 constraints. The same constrained problem solved three times, differing only in which basis Z is used for the null space of A. The orthonormal basis, from a QR of Aᵀ, has κ(Z) = 1 exactly and is dense — 100 per cent of its entries are nonzero. The fundamental basis built on the first 4 columns has κ(Z) = 1.994·10⁸, and its reduced Hessian comes out at 3.801·10¹⁶, which is κ(Z)² to within a factor of 0.956 — the square attained rather than bounded. Its answer is wrong by 0.05179, against 1.07·10⁻¹⁵ for the orthonormal one. Choosing the same kind of basis by pivoting instead gives κ(Z) = 2.06, an error of 6.71·10⁻¹⁶, and the same 50 per cent density: all of the sparsity and none of the loss.

null-basis-bars is one function in lib/figures/nullbasis.js — the basis of a constraint — three spans of one null space, and the square between them. 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 same constrained problem solved three times, differing only in which basis Z is used for the null space of A. The orthonormal basis, from a QR of Aᵀ, has κ(Z) = 1 exactly and is dense — 100 per cent of its entries are nonzero. The fundamental basis built on the first 4 columns has κ(Z) = 1.994·10⁸, and its reduced Hessian comes out at 3.801·10¹⁶, which is κ(Z)² to within a factor of 0.956 — the square attained rather than bounded. Its answer is wrong by 0.05179, against 1.07·10⁻¹⁵ for the orthonormal one. Choosing the same kind of basis by pivoting instead gives κ(Z) = 2.06, an error of 6.71·10⁻¹⁶, and the same 50 per cent density: all of the sparsity and none of the loss.

The same constrained problem solved three times, differing only in which basis Z is used for the null space of A. The orthonormal basis, from a QR of Aᵀ, has κ(Z) = 1 exactly and is dense — 100 per cent of its entries are nonzero. The fundamental basis built on the first 4 columns has κ(Z) = 4.436, and its reduced Hessian comes out at 83.77, which is κ(Z)² to within a factor of 4.26 — the square attained rather than bounded. Its answer is wrong by 1.26·10⁻¹⁵, against 1.29·10⁻¹⁵ for the orthonormal one. Choosing the same kind of basis by pivoting instead gives κ(Z) = 2.258, an error of 1.41·10⁻¹⁵, and the same 50 per cent density: all of the sparsity and none of the loss.

The same constrained problem solved three times, differing only in which basis Z is used for the null space of A. The orthonormal basis, from a QR of Aᵀ, has κ(Z) = 1 exactly and is dense — 100 per cent of its entries are nonzero. The fundamental basis built on the first 4 columns has κ(Z) = 1.994·10⁸, and its reduced Hessian comes out at 3.801·10¹⁶, which is κ(Z)² to within a factor of 0.956 — the square attained rather than bounded. Its answer is wrong by 0.05179, against 1.07·10⁻¹⁵ for the orthonormal one. Choosing the same kind of basis by pivoting instead gives κ(Z) = 2.06, an error of 6.71·10⁻¹⁶, and the same 50 per cent density: all of the sparsity and none of the loss.

The same constrained problem solved three times, differing only in which basis Z is used for the null space of A. The orthonormal basis, from a QR of Aᵀ, has κ(Z) = 1 exactly and is dense — 100 per cent of its entries are nonzero. The fundamental basis built on the first 4 columns has κ(Z) = 1.997·10⁴, and its reduced Hessian comes out at 5.845·10⁸, which is κ(Z)² to within a factor of 1.47 — the square attained rather than bounded. Its answer is wrong by 8.504·10⁻¹⁰, against 2.25·10⁻¹⁵ for the orthonormal one. Choosing the same kind of basis by pivoting instead gives κ(Z) = 2.06, an error of 1.33·10⁻¹⁵, and the same 50 per cent density: all of the sparsity and none of the loss.

The same constrained problem solved three times, differing only in which basis Z is used for the null space of A. The orthonormal basis, from a QR of Aᵀ, has κ(Z) = 1 exactly and is dense — 100 per cent of its entries are nonzero. The fundamental basis built on the first 4 columns has κ(Z) = 205.7, and its reduced Hessian comes out at 8.39·10⁴, which is κ(Z)² to within a factor of 1.98 — the square attained rather than bounded. Its answer is wrong by 3.074·10⁻¹³, against 1.15·10⁻¹⁵ for the orthonormal one. Choosing the same kind of basis by pivoting instead gives κ(Z) = 2.061, an error of 2.88·10⁻¹⁶, and the same 50 per cent density: all of the sparsity and none of the loss.

The same constrained problem solved three times, differing only in which basis Z is used for the null space of A. The orthonormal basis, from a QR of Aᵀ, has κ(Z) = 1 exactly and is dense — 100 per cent of its entries are nonzero. The fundamental basis built on the first 4 columns has κ(Z) = 1.994·10¹⁰, and its reduced Hessian comes out at 4.871·10¹⁸, which is κ(Z)² to within a factor of 0.0123 — the square attained rather than bounded. Its answer is wrong by 0.3173, against 8.67·10⁻¹⁶ for the orthonormal one. Choosing the same kind of basis by pivoting instead gives κ(Z) = 2.06, an error of 8.07·10⁻¹⁶, and the same 50 per cent density: all of the sparsity and none of the loss.

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.

12 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 basisComparison, 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 165 of 306 generators — 150 print a residual and 15 are exempt with a published reason; 141 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.