Two bases for one subspace, at an interpolation point of 20

At its defaults it draws two bases for one subspace, at an interpolation point of 20. κ₂ of the moment basis (σI − A)⁻¹B, (σI − A)⁻²B, … at a single point σ = 20, against κ₂ of the same number of solves spent at points spread across the model's poles and orthogonalised as they are built. The moment basis runs 10.9 to 7.75·10⁹, multiplying by about 65.8 per vector, while the spread one stays at 1. The reason is the one behind every Krylov basis on this site: repeated application of one operator converges to a direction, so every new moment is nearly the last. The rate is set by the ratio of distances from σ to the poles, which means the user's own choice of interpolation point decides how fast their basis becomes unusable — and spending the solves at different points instead is both a better approximation and a basis that survives.

moment-basis is one function in lib/figures/ratkrylov.js — matching the function where you choose — interpolation points, the basis they need, and the fixed point. 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.

κ₂ of the moment basis (σI − A)⁻¹B, (σI − A)⁻²B, … at a single point σ = 20, against κ₂ of the same number of solves spent at points spread across the model's poles and orthogonalised as they are built. The moment basis runs 10.9 to 7.75·10⁹, multiplying by about 65.8 per vector, while the spread one stays at 1. The reason is the one behind every Krylov basis on this site: repeated application of one operator converges to a direction, so every new moment is nearly the last. The rate is set by the ratio of distances from σ to the poles, which means the user's own choice of interpolation point decides how fast their basis becomes unusable — and spending the solves at different points instead is both a better approximation and a basis that survives.

κ₂ of the moment basis (σI − A)⁻¹B, (σI − A)⁻²B, … at a single point σ = 20, against κ₂ of the same number of solves spent at points spread across the model's poles and orthogonalised as they are built. The moment basis runs 10.9 to 7.75·10⁹, multiplying by about 65.8 per vector, while the spread one stays at 1. The reason is the one behind every Krylov basis on this site: repeated application of one operator converges to a direction, so every new moment is nearly the last. The rate is set by the ratio of distances from σ to the poles, which means the user's own choice of interpolation point decides how fast their basis becomes unusable — and spending the solves at different points instead is both a better approximation and a basis that survives.

κ₂ of the moment basis (σI − A)⁻¹B, (σI − A)⁻²B, … at a single point σ = 0.5, against κ₂ of the same number of solves spent at points spread across the model's poles and orthogonalised as they are built. The moment basis runs 14.9 to 2.55·10¹², multiplying by about 183 per vector, while the spread one stays at 1. The reason is the one behind every Krylov basis on this site: repeated application of one operator converges to a direction, so every new moment is nearly the last. The rate is set by the ratio of distances from σ to the poles, which means the user's own choice of interpolation point decides how fast their basis becomes unusable — and spending the solves at different points instead is both a better approximation and a basis that survives.

κ₂ of the moment basis (σI − A)⁻¹B, (σI − A)⁻²B, … at a single point σ = 3, against κ₂ of the same number of solves spent at points spread across the model's poles and orthogonalised as they are built. The moment basis runs 13.5 to 7.08·10¹¹, multiplying by about 148 per vector, while the spread one stays at 1. The reason is the one behind every Krylov basis on this site: repeated application of one operator converges to a direction, so every new moment is nearly the last. The rate is set by the ratio of distances from σ to the poles, which means the user's own choice of interpolation point decides how fast their basis becomes unusable — and spending the solves at different points instead is both a better approximation and a basis that survives.

κ₂ of the moment basis (σI − A)⁻¹B, (σI − A)⁻²B, … at a single point σ = 120, against κ₂ of the same number of solves spent at points spread across the model's poles and orthogonalised as they are built. The moment basis runs 14.6 to 5.04·10⁷, multiplying by about 18.5 per vector, while the spread one stays at 1. The reason is the one behind every Krylov basis on this site: repeated application of one operator converges to a direction, so every new moment is nearly the last. The rate is set by the ratio of distances from σ to the poles, which means the user's own choice of interpolation point decides how fast their basis becomes unusable — and spending the solves at different points instead is both a better approximation and a basis that survives.

κ₂ of the moment basis (σI − A)⁻¹B, (σI − A)⁻²B, … at a single point σ = 800, against κ₂ of the same number of solves spent at points spread across the model's poles and orthogonalised as they are built. The moment basis runs 38.9 to 3.51·10⁸, multiplying by about 16.4 per vector, while the spread one stays at 1. The reason is the one behind every Krylov basis on this site: repeated application of one operator converges to a direction, so every new moment is nearly the last. The rate is set by the ratio of distances from σ to the poles, which means the user's own choice of interpolation point decides how fast their basis becomes unusable — and spending the solves at different points instead is both a better approximation and a basis that survives.

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.

8 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 momentBasis, rationalBasis, 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.