MINRES on a saddle-point system under three Schur-complement approximations, and under none

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.

For each of twenty-four systems — the constraint's condition number 10, 100, 1,000 or 10,000, the Hessian's 10, 100 or 10,000, two seeds each, every one with a Hessian and a constraint of norm one — the γ on a quarter-decade grid from 0.01 to 10⁸ at which MINRES steps times conjugate gradient steps per inner solve is least. They run from 0.0316 to 3.16·10⁵. at a constraint condition number of 10 they run from 0.0316 to 1.78; at a constraint condition number of 100 they run from 0.562 to 100; at a constraint condition number of 1000 they run from 56.2 to 1778; at a constraint condition number of 10⁴ they run from 5623 to 3.16·10⁵.

For each Hessian condition number and seed, MINRES steps times conjugate gradient steps per inner solve, divided by the least over the γ grid, against γ on a logarithmic axis, capped at 2.6. The least work sits at γ = 1000, 1778, 1000, 1778, 100, 56.2. A filled dot marks where the rule "the smallest γ at which the smallest generalised eigenvalue reaches 0.03" lands: at 1.20, 1.09, 1.06, 1.09, 1.05, 1.10 times the least work.

For the same twenty-four systems, the smallest of γs over one plus γs, over the eigenvalues s of the Schur complement, at the γ of least work, against the constraint's condition number on a logarithmic axis. It runs from 0.0020 to 0.0590 — a factor of 30. The dashed line is 0.03, the target the rule below uses.

For each target c, the rule takes the smallest γ on the grid at which the smallest generalised eigenvalue reaches c, and its work is divided by that system's least; the median and the worst over twenty-four systems are drawn against c on a logarithmic axis. at 0.003: median 1.314, worst 1.909; at 0.01: median 1.100, worst 1.545; at 0.03: median 1.067, worst 1.286; at 0.06: median 1.150, worst 1.348; at 0.1: median 1.200, worst 1.400; at 0.3: median 1.270, worst 1.600. The best single γ for every system, 31.6, has a median of 1.364 and a worst of 1.612.

One dot per system: horizontally the smallest forward error of the answer, against the exact rational solution, over the whole γ grid; vertically the forward error at the γ the rule chooses. On the dashed diagonal the rule gives up nothing. The median system is 11.6 times its best and the worst 4561 times; the largest error the rule accepts is 4.38·10⁻¹⁰.

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.

26 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 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.