mgw-convergence

At its defaults it draws minres on a saddle-point system under three schur-complement approximations, and under none. The same system at 12 unknowns and 5 constraints, solved four ways. With the exact Schur complement the preconditioned matrix has three distinct eigenvalues and the residual falls to 4.109·10⁻¹⁵ in three steps, after which nothing is left to remove. Replacing S by A diag(H)⁻¹Aᵀ costs 11 steps and replacing it by a scaled AAᵀ costs 11; the unpreconditioned system takes 22. The exact preconditioner is unaffordable — forming S costs 5 solves with H and a decomposition — so its value is as the statement the cheap ones are measured against, and the measurement needs no reference solution: the distance from {1 − φ, 1, φ} is a property of the approximation alone.

mgw-convergence is one function in lib/figures/blockprec.js — preconditioning a saddle point — three eigenvalues, and a spectrum with no constraint in it. 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 system at 12 unknowns and 5 constraints, solved four ways. With the exact Schur complement the preconditioned matrix has three distinct eigenvalues and the residual falls to 4.109·10⁻¹⁵ in three steps, after which nothing is left to remove. Replacing S by A diag(H)⁻¹Aᵀ costs 11 steps and replacing it by a scaled AAᵀ costs 11; the unpreconditioned system takes 22. The exact preconditioner is unaffordable — forming S costs 5 solves with H and a decomposition — so its value is as the statement the cheap ones are measured against, and the measurement needs no reference solution: the distance from {1 − φ, 1, φ} is a property of the approximation alone.

The same system at 12 unknowns and 5 constraints, solved four ways. With the exact Schur complement the preconditioned matrix has three distinct eigenvalues and the residual falls to 2.947·10⁻¹⁴ in three steps, after which nothing is left to remove. Replacing S by A diag(H)⁻¹Aᵀ costs 13 steps and replacing it by a scaled AAᵀ costs 13; the unpreconditioned system takes 25. The exact preconditioner is unaffordable — forming S costs 5 solves with H and a decomposition — so its value is as the statement the cheap ones are measured against, and the measurement needs no reference solution: the distance from {1 − φ, 1, φ} is a property of the approximation alone.

The same system at 12 unknowns and 5 constraints, solved four ways. With the exact Schur complement the preconditioned matrix has three distinct eigenvalues and the residual falls to 6.534·10⁻¹⁶ in three steps, after which nothing is left to remove. Replacing S by A diag(H)⁻¹Aᵀ costs 3 steps and replacing it by a scaled AAᵀ costs 3; the unpreconditioned system takes 13. The exact preconditioner is unaffordable — forming S costs 5 solves with H and a decomposition — so its value is as the statement the cheap ones are measured against, and the measurement needs no reference solution: the distance from {1 − φ, 1, φ} is a property of the approximation alone.

The same system at 12 unknowns and 3 constraints, solved four ways. With the exact Schur complement the preconditioned matrix has three distinct eigenvalues and the residual falls to 4.31·10⁻¹⁵ in three steps, after which nothing is left to remove. Replacing S by A diag(H)⁻¹Aᵀ costs 7 steps and replacing it by a scaled AAᵀ costs 7; the unpreconditioned system takes 18. The exact preconditioner is unaffordable — forming S costs 3 solves with H and a decomposition — so its value is as the statement the cheap ones are measured against, and the measurement needs no reference solution: the distance from {1 − φ, 1, φ} is a property of the approximation alone.

The same system at 12 unknowns and 5 constraints, solved four ways. With the exact Schur complement the preconditioned matrix has three distinct eigenvalues and the residual falls to 4.109·10⁻¹⁵ in three steps, after which nothing is left to remove. Replacing S by A diag(H)⁻¹Aᵀ costs 11 steps and replacing it by a scaled AAᵀ costs 11; the unpreconditioned system takes 22. The exact preconditioner is unaffordable — forming S costs 5 solves with H and a decomposition — so its value is as the statement the cheap ones are measured against, and the measurement needs no reference solution: the distance from {1 − φ, 1, φ} is a property of the approximation alone.

The same system at 12 unknowns and 5 constraints, solved four ways. With the exact Schur complement the preconditioned matrix has three distinct eigenvalues and the residual falls to 5.317·10⁻¹⁰ in three steps, after which nothing is left to remove. Replacing S by A diag(H)⁻¹Aᵀ costs 15 steps and replacing it by a scaled AAᵀ costs 17; the unpreconditioned system takes 42. The exact preconditioner is unaffordable — forming S costs 5 solves with H and a decomposition — so its value is as the statement the cheap ones are measured against, and the measurement needs no reference solution: the distance from {1 − φ, 1, φ} is a property of the approximation alone.

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.

10 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 minres, blockDiagonalRun, 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.