Singular values and |uₖᵀb|, with and without 0.10% noise

At its defaults it draws singular values and |uₖᵀb|, with and without 0.10% noise. Three curves against the index on a logarithmic vertical axis. The singular values fall exponentially to the level of rounding. With an exact right-hand side the coefficients fall faster and every term of the solution stays bounded. With noise they flatten at index 32, and from there on each term is noise divided by a σ of 6.8·10⁻⁴.

picard-plot is one function in lib/figures/regular.js — regularisation — the filter, the corner, and the answer that is a choice. 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.

Three curves against the index on a logarithmic vertical axis. The singular values fall exponentially to the level of rounding. With an exact right-hand side the coefficients fall faster and every term of the solution stays bounded. With noise they flatten at index 32, and from there on each term is noise divided by a σ of 6.8·10⁻⁴.

The Picard picture for noise whose samples are correlated with coefficient 0.9. The singular values fall to rounding; the exact coefficients fall faster; the noisy ones flatten. The noise's own coefficients are drawn too, smoothed, and 98% of its energy sits in the first 28 directions, tilting its floor by 1.14 decades from the first eight to the last sixteen. The crossing is at 43 and the best truncation at 31.

The Picard picture for noise whose samples are correlated with coefficient 0. The singular values fall to rounding; the exact coefficients fall faster; the noisy ones flatten. The noise's own coefficients are drawn too, smoothed, and 43% of its energy sits in the first 28 directions, tilting its floor by 0.07 decades from the first eight to the last sixteen. The crossing is at 32 and the best truncation at 28.

Each dot is one noise draw: the error a rule's choice of λ produced, divided by the least error any λ produced on that draw. Generalised cross-validation more than doubles the best error on 3 of 48 draws, the discrepancy principle on 0, the L-curve corner on 8. Median costs are 1.005, 1.063 and 1.506.

Each dot is one noise draw: the error a rule's choice of λ produced, divided by the least error any λ produced on that draw. Generalised cross-validation more than doubles the best error on 14 of 48 draws, the discrepancy principle on 0, the L-curve corner on 22. Median costs are 1.027, 1.187 and 1.947.

For each correlation from 0 to 0.99, the number of 48 noise draws on which a rule's λ produced more than twice the least error available. Generalised cross-validation: 3, 4, 6, 10, 14, 11; after whitening: 3, 3, 2, 2, 3, 3. The L-curve corner: 8, 12, 19, 19, 22, 28. The discrepancy principle doubles it on none, while its median cost rises from 1.063 to 1.220.

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.

12 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.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

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.