Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Formatted arithmetic — where it appears

Arithmetic on matrices held in a compressed format, where every operation returns a result in that same format. Addition is the operation such a format is not closed under, so each sum is followed by a truncation back to a smaller rank, and the accuracy of a computed factor is set by that tolerance rather than by the machine's precision.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

0816243211.11.21.31.4terms added, each followed by a truncationerror ⁄ best rank-k erroroptimalterms with nothing in commona subspace that driftsthe rounding nobody should have feareddrifting, worst excess1independent, worst excess1a linear bound would say32energy discarded, first1.8·10⁻⁷energy discarded, last0.03thirty-two roundingsand four per cent

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 10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²ε the blocks were compressed at‖A − LLᵀ‖ ⁄ ‖A‖the factorisationthe knob, on a factorisationtruncations34residual at 10⁻⁴2.2·10⁻⁵residual at 10⁻¹⁰9.9·10⁻¹²slope1worst ratio to the representation1.8ten decadesand a slope of one

A knob calibrated in residuals

A formatted Cholesky has two numbers in it and only one of them is an accuracy. Across twelve trees — three sizes by four leaf sizes — the leaf moves the truncation count from 0 to 258 and moves the ranks of the blocks not at all, while the residual follows the tolerance at slopes between 1.022 and 1.046 and sits at about a tenth of it throughout.

hierarchy · Recompression 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 count that is not the budget

A Cholesky performed inside a low-rank format truncates 0, 2, 10, 34 and 98 times as the leaf falls from 128 to 8, and those five integers are the same at every accuracy from 10⁻¹² to 10⁻². Across all ten decades the factorisation's residual stays below the representation's own error at a ratio between 0.81 and 1.00 — with two entries that read 1.83 and 1.78, and neither of them is accumulation.

hierarchy · Recompression

Named alongside it

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

Cholesky factorisationHierarchical matrixLow-rank approximationRecompressionResidualTruncated SVDAdmissibilityEckart–YoungError accumulationBackward errorOff-diagonal rankRandom walk

All concepts