Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Recompression — where it appears

Truncating a low-rank representation back to a smaller rank after an operation has grown it. It is unavoidable in any format that is not closed under its own arithmetic, and what it costs is measured rather than assumed.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

02040608010010⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸truncations performed by the factorisationrelative errorthe representation's own error‖A − LLᵀ‖ ⁄ ‖A‖no decomposition without its residualresidual, 0 truncations2.1·10⁻¹⁰residual, 98 truncations1.1·10⁻⁹representation, deepest1.4·10⁻⁹residual ⁄ representation0.81levels, deepest5a hundred approximate stepsand an exact-looking factorisation

The rounding that was not the problem

A rank-k block plus a rank-k block is a rank-2k block, exactly, so every arithmetic in this format truncates after every addition. A Cholesky performed inside it does ninety-eight of those and its residual is 1.14·10⁻⁹ against a representation error of 1.40·10⁻⁹ — the roundings cost nothing measurable.

hierarchy · recompression 081624324010⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²conjugate gradient steprelative residualabove, on their own scale: rank asked 5, rank kept 3solid: the budgeted residual · light: no budgetbudget 3rank asked for5rank kept3residual, budgeted3.8·10⁻⁴residual, unbudgeted1.6·10⁻¹⁴numbers stored150the step asks for moreat every step

An iterate that must be made smaller

Applying a Kronecker-sum operator to a low-rank iterate multiplies its ranks by d and adding two of them adds their ranks, so a solver in a compressed format cannot keep what it produces. Every step is followed by a truncation — and whether that truncation is a floor on the residual depends on the right-hand side rather than on the truncation.

iterative · low rank iteration

Named alongside it

The objects these essays reach for when they reach for this one.

Low-rank approximationResidualCholesky factorisationConjugate gradientsCurse of dimensionalityEckart–YoungError accumulationFormatted arithmeticHierarchical matrixKronecker sumKrylov subspaceModel problem

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