Three residuals, and the one a solver returns is the one about nothing

At its defaults it draws three residuals, and the one a solver returns is the one about nothing. For a rational approximant with m poles, on a target set reaching to 0.3: ‖T̃(λ)x‖ — the residual against the problem the eigensolver was handed — stays at 1.2·10⁻¹⁵ to 1.5·10⁻¹² — it does not fall with m, and past a point it slowly rises, because each added pole makes the fit's own basis worse conditioned; ‖T(λ)x‖, which costs one further evaluation of γ√(λ + c), falls with the approximation; and the forward error against the closed form falls with it, staying 20.5 to 24.8 times larger. The free number rises by 1306 across the sweep while the answer improves by 885. So the free number says nothing and the nearly-free number says almost everything. Evaluate the residual against the function you asked about, not against the one you handed over is the whole practical content of this field.

three-residuals is one function in lib/figures/ratapprox.js — the problem the solver was given — the approximation committed before the arithmetic. 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.

For a rational approximant with m poles, on a target set reaching to 0.3: ‖T̃(λ)x‖ — the residual against the problem the eigensolver was handed — stays at 1.2·10⁻¹⁵ to 1.5·10⁻¹² — it does not fall with m, and past a point it slowly rises, because each added pole makes the fit's own basis worse conditioned; ‖T(λ)x‖, which costs one further evaluation of γ√(λ + c), falls with the approximation; and the forward error against the closed form falls with it, staying 20.5 to 24.8 times larger. The free number rises by 1306 across the sweep while the answer improves by 885. So the free number says nothing and the nearly-free number says almost everything. Evaluate the residual against the function you asked about, not against the one you handed over is the whole practical content of this field.

For a rational approximant with m poles, on a target set reaching to 0.3: ‖T̃(λ)x‖ — the residual against the problem the eigensolver was handed — stays at 1.2·10⁻¹⁵ to 1.5·10⁻¹² — it does not fall with m, and past a point it slowly rises, because each added pole makes the fit's own basis worse conditioned; ‖T(λ)x‖, which costs one further evaluation of γ√(λ + c), falls with the approximation; and the forward error against the closed form falls with it, staying 20.5 to 24.8 times larger. The free number rises by 1306 across the sweep while the answer improves by 885. So the free number says nothing and the nearly-free number says almost everything. Evaluate the residual against the function you asked about, not against the one you handed over is the whole practical content of this field.

For a rational approximant with m poles, on a target set reaching to 1: ‖T̃(λ)x‖ — the residual against the problem the eigensolver was handed — stays at 1.8·10⁻¹⁵ to 2·10⁻⁸ — it does not fall with m, and past a point it slowly rises, because each added pole makes the fit's own basis worse conditioned; ‖T(λ)x‖, which costs one further evaluation of γ√(λ + c), falls with the approximation; and the forward error against the closed form falls with it, staying 19.5 to 21.2 times larger. The free number rises by 1.14·10⁷ across the sweep while the answer improves by 2.79·10⁴. So the free number says nothing and the nearly-free number says almost everything. Evaluate the residual against the function you asked about, not against the one you handed over is the whole practical content of this field.

For a rational approximant with m poles, on a target set reaching to 0.7: ‖T̃(λ)x‖ — the residual against the problem the eigensolver was handed — stays at 1.1·10⁻¹⁵ to 10·10⁻¹⁰ — it does not fall with m, and past a point it slowly rises, because each added pole makes the fit's own basis worse conditioned; ‖T(λ)x‖, which costs one further evaluation of γ√(λ + c), falls with the approximation; and the forward error against the closed form falls with it, staying 19.6 to 24.8 times larger. The free number rises by 8.72·10⁵ across the sweep while the answer improves by 8634. So the free number says nothing and the nearly-free number says almost everything. Evaluate the residual against the function you asked about, not against the one you handed over is the whole practical content of this field.

For a rational approximant with m poles, on a target set reaching to 0.5: ‖T̃(λ)x‖ — the residual against the problem the eigensolver was handed — stays at 1.2·10⁻¹⁵ to 7.1·10⁻¹¹ — it does not fall with m, and past a point it slowly rises, because each added pole makes the fit's own basis worse conditioned; ‖T(λ)x‖, which costs one further evaluation of γ√(λ + c), falls with the approximation; and the forward error against the closed form falls with it, staying 21.2 to 24.8 times larger. The free number rises by 6.05·10⁴ across the sweep while the answer improves by 2396. So the free number says nothing and the nearly-free number says almost everything. Evaluate the residual against the function you asked about, not against the one you handed over is the whole practical content of this field.

For a rational approximant with m poles, on a target set reaching to 0.4: ‖T̃(λ)x‖ — the residual against the problem the eigensolver was handed — stays at 2.4·10⁻¹⁵ to 3.3·10⁻¹¹ — it does not fall with m, and past a point it slowly rises, because each added pole makes the fit's own basis worse conditioned; ‖T(λ)x‖, which costs one further evaluation of γ√(λ + c), falls with the approximation; and the forward error against the closed form falls with it, staying 21.5 to 24.8 times larger. The free number rises by 1.4·10⁴ across the sweep while the answer improves by 1456. So the free number says nothing and the nearly-free number says almost everything. Evaluate the residual against the function you asked about, not against the one you handed over is the whole practical content of this field.

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 fitRational, rationalEigen, residualsAt, 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.