Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Drop tolerance — where it appears

The constant an incomplete factorisation compares an entry against before deciding to discard it. An entry is dropped when its magnitude falls below the tolerance times the geometric mean of the two diagonal entries it sits between, so one number fixes both the sparsity of the factor and the residual of the factorisation.

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

015304560759010512010⁻²10⁻¹110¹steprelative sizeleast error: 20discrepancy stop: 7errorresidualthe knob is an integerleast error, at step20error there0.14error at step 1206the residual falls at every stepthe error turns and keeps rising

The zero you are allowed to write

A deflation criterion sets a subdiagonal entry to zero because it is small. A drop tolerance discards an entry of a factor because it is small. A truncation discards a singular value because it is small. Three fields, three vocabularies, no shared arithmetic — and plotted as work saved against error accepted, one curve.

error · Deliberate zero 10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²-1-0.500.51relative backward error acceptedfraction of the work not doneQR deflation criterionincomplete Cholesky droplow-rank truncationthree fields, no shared arithmeticslope, deflation criterion0.039slope, drop tolerance0.051slope, rank truncation0.23widest apart, as a ratio5.7a tolerance is an offerand the three offers are one offer

A tolerance is priced by the problem

Three tolerances from three fields sit on one pair of axes and agree to within a factor of 5.74. That factor is the ratio of the two curves that cannot move. Change the only problem in the comparison and the third curve's fitted slope swings from 0.188 to 0.040 while the printed spread does not shift by a digit.

error · Deliberate zero

Named alongside it

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

Backward errorDeflationEckart–YoungIncomplete factorisationLow-rank approximationPreconditioningToleranceFill-inModel problemThe QR algorithm

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