Filter factors — where it appears
Also named here as truncated svd — the same set of essays touches all of them, so they are one junction rather than several.
When the answer is a choice
A backward-stable least-squares solve of this problem returns an answer whose relative error is 5.5·10⁸. Nothing went wrong. The singular values decay exponentially with no gap anywhere in them, the data does not determine the answer, and something outside the data has to choose — which is the computation rather than a preliminary to it.
Where the answer stops being in the data
The Picard condition finds the index where a noisy right-hand side stops carrying signal, from the data alone, with no knowledge of the answer. It lands at 32 where the truncation that actually minimises the error is 28 — and at 45 where the best is 38. It overshoots at every noise level from 1% to 0.001%, and it overshoots for a reason.
Named alongside it
The objects these essays reach for when they reach for this one.
Ill posed problemRegularisationSingular value decompositionTruncated svdCondition numberDeconvolutionNoise floorPicard conditionSignal to noiseTikhonov regularisation