Steihaug's model decrease as a share of the exact trust-region answer, against the radius

At its defaults it draws steihaug's model decrease as a share of the exact trust-region answer, against the radius. The quadratic model is ½xᵀAx − bᵀx on the same 40×40 matrix, positive definite apart from one eigenvalue at -0.1. At Δ = 0.25 the truncated conjugate gradient gets 89.2 per cent of what the exact subproblem solution gets, for 2 matrix–vector products against a full symmetric eigendecomposition and a secular equation. The share falls monotonically to 36.9 per cent at Δ = 16. The open marks are the radii at which the step stopped because the curvature turned rather than because it reached the boundary; the crossover is at Δ = 4.

trust-region-share is one function in lib/figures/curvature.js — negative curvature — the division that cannot be done, as an output. 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 quadratic model is ½xᵀAx − bᵀx on the same 40×40 matrix, positive definite apart from one eigenvalue at -0.1. At Δ = 0.25 the truncated conjugate gradient gets 89.2 per cent of what the exact subproblem solution gets, for 2 matrix–vector products against a full symmetric eigendecomposition and a secular equation. The share falls monotonically to 36.9 per cent at Δ = 16. The open marks are the radii at which the step stopped because the curvature turned rather than because it reached the boundary; the crossover is at Δ = 4.

The quadratic model is ½xᵀAx − bᵀx on the same 60×60 matrix, positive definite apart from one eigenvalue at -0.1. At Δ = 0.25 the truncated conjugate gradient gets 94.9 per cent of what the exact subproblem solution gets, for 3 matrix–vector products against a full symmetric eigendecomposition and a secular equation. The share falls monotonically to 30.1 per cent at Δ = 16. The open marks are the radii at which the step stopped because the curvature turned rather than because it reached the boundary; the crossover is at Δ = 2.

The quadratic model is ½xᵀAx − bᵀx on the same 40×40 matrix, positive definite apart from one eigenvalue at -0.01. At Δ = 0.25 the truncated conjugate gradient gets 89.7 per cent of what the exact subproblem solution gets, for 2 matrix–vector products against a full symmetric eigendecomposition and a secular equation. The share falls monotonically to 40.8 per cent at Δ = 16. The open marks are the radii at which the step stopped because the curvature turned rather than because it reached the boundary; the crossover is at Δ = —.

The quadratic model is ½xᵀAx − bᵀx on the same 40×40 matrix, positive definite apart from one eigenvalue at -10. At Δ = 0.25 the truncated conjugate gradient gets 39.6 per cent of what the exact subproblem solution gets, for 2 matrix–vector products against a full symmetric eigendecomposition and a secular equation. The share falls monotonically to 35.2 per cent at Δ = 16. The open marks are the radii at which the step stopped because the curvature turned rather than because it reached the boundary; the crossover is at Δ = 0.25.

The quadratic model is ½xᵀAx − bᵀx on the same 40×40 matrix, positive definite apart from one eigenvalue at -1. At Δ = 0.25 the truncated conjugate gradient gets 83.2 per cent of what the exact subproblem solution gets, for 2 matrix–vector products against a full symmetric eigendecomposition and a secular equation. The share falls monotonically to 29.3 per cent at Δ = 16. The open marks are the radii at which the step stopped because the curvature turned rather than because it reached the boundary; the crossover is at Δ = 1.

The quadratic model is ½xᵀAx − bᵀx on the same 40×40 matrix, positive definite apart from one eigenvalue at -0.001. At Δ = 0.25 the truncated conjugate gradient gets 89.8 per cent of what the exact subproblem solution gets, for 2 matrix–vector products against a full symmetric eigendecomposition and a secular equation. The share falls monotonically to 68.3 per cent at Δ = 16. The open marks are the radii at which the step stopped because the curvature turned rather than because it reached the boundary; the crossover is at Δ = —.

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.

9 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 trustRegionSweep, 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.