Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Threshold pivoting — where it appears

Accepting a pivot that is within a stated factor of the largest available, rather than insisting on the largest. It trades stability against fill in a sparse factorisation, and the two resources are not in the same units.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

the matrix105 entriescorner first — sparsest227 entries, growth 1.9·10¹¹largest first — safe242 entries, growth 1.19the middle factor is the smaller one, and its answer has no correct digitsboth factorisations reproduce the matrix‖PA − LU‖/‖A‖, sparsest3.8·10⁻¹⁷‖PA − LU‖/‖A‖, pivoted5.4·10⁻¹⁷forward error, sparsest3·10⁻⁵forward error, pivoted4.8·10⁻¹⁶red marks are entries elimination createdthe fill argument and the stability argument disagree

Structure and stability stop being separable

The sparsest variable to eliminate on this matrix has a diagonal entry of 10⁻¹². Eliminating it produces the smaller factor, reproduces the matrix to 3.8·10⁻¹⁷ — better than pivoting does — and returns an answer wrong in the fifth digit.

sparsity · Sparse pivoting 10⁻³10⁻²10⁻¹1110¹10²10³10⁴pivot threshold τgrowth factor · entries in L+U, ÷ entries in Agrowthfillthe library default‖PA − LU‖/‖A‖ at τ = 0.12.9·10⁻¹⁶growth at τ = 0.138entries in L+U at τ = 0.1372one knob, two measurements, opposite directionsand the default is most of both

A threshold between fill and growth

One number decides how small a pivot an elimination will accept. At 0.001 the factor holds 172 entries and the matrix grows by 1,330; at 1 it holds 260 and grows by 1.2. The libraries ship 0.1, and the measurement says why.

sparsity · Sparse pivoting 02468101210⁻¹⁸10⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸member of the sequencebackward error of the solvefresh order · equilibrated kept orderringed: another pivot replaced hereone order, two sets of unitsfresh order, last member9.5·10⁻¹⁷kept order, last member4.8·10⁻⁹equilibrated, last member6.5·10⁻¹⁷pivots replaced, kept2pivots replaced, equilibrated0the perturbation floor √u1.1·10⁻⁸a reused pivot order is safe or notdepending on what the rows are measured in

The order that was right last time

A pivot order computed once and reused across a sequence saves the symbolic phase, and the price is that a pivot which was large may now be small. Replacing it with √u·‖A‖ costs eight orders of backward error and iterative refinement recovers a factor of 8.8 of them. Divide each row by its largest entry first and the same reuse costs nothing at all.

sparsity · Sparse pivoting 10⁻³10⁻²10⁻¹1110¹10²10³10⁴pivot threshold τgrowth factor · entries in L + U ÷ entries in Agrowth, rows onlygrowth, rows and columnsfill ratio, rows onlyfill ratio, rows and columns8×8 grid, 64 unknowns, 288 entriesτ = 0.001: rows only — 741 entries; growth2209τ = 0.001: rows and columns — 640 entries; growth2.5τ = 0.1: rows only — 875 entries; growth38τ = 0.1: rows and columns — 640 entries; growth2.5τ = 1: rows only — 986 entries; growth1.2τ = 1: rows and columns — 659 entries; growth1.2dashed: a fixed column · solid: the column chosen toothe same threshold rule in both

The column that was never fixed

Every threshold-pivoting measurement so far chose the pivot row in a fixed column, and the routine's own description said that choosing the column as well would change the constants and not the argument. Measured, it changes the argument. On the 8×8 conflict grid the factor shrinks from 875 entries to 640 at the library default, and the growth factor that climbed to 2,209 as the threshold loosened stays at 2.54 at every threshold from 0.3 down to 0.001. What does most of the work is not the column but which of several equally cheap entries is taken — and on random sparse matrices, choosing the column without that makes the growth worse.

sparsity · Sparse pivoting 80 matrices, τ = 0.1, ties to the largestno column search, median fill244one column136every column109share of the gain at one column0.804896144192240columns the search may look atmedian fill123468121624allno column search at allmedian fillmedian growthworst growth — see the captionthe fill is the quantity the width buysthe worst case is the draw

How few columns the search needs

A full row-and-column pivot search is quadratic in the active submatrix at every step, and no library performs one. Looking at a single sparsest column takes the median fill from 244 to 136 where the full search reaches 109 — four fifths of the benefit for a linear scan — and that share is 79, 83, 80, 89 and 92 per cent across five thresholds. The worst growth appears to favour the narrow search by a factor of six, and on the next draw it favours the wide one by two.

sparsity · Sparse pivoting a 28 × 28 saddle-point matrixnatural order, entries113sparsest order, entries72two-by-two pivots, loosest test0and at the strictest3023466992115the constant in the pivot testentries in the factor0.10.30.640.80.95133natural ordersparsest ordergrowth, naturalgrowth, sparsestnumbers above the points: two-by-two pivots takenboth curves favour the sparser order

The freedom a symmetric factorisation does not have

Permuting rows and columns together leaves no column to choose, so the conflict between the sparsest pivot and the sound one should be worse rather than better. On a saddle-point matrix whose constraint rows have no diagonal entry at all, it is not there: taking the sparsest available pivot holds 70 entries against the natural order's 113 and a growth of 1.28 against 1.83 — better on both currencies at once, at every setting of the pivot test. The two-by-two blocks that make it legal cost 1.33 entries apiece.

sparsity · Sparse pivoting noise that halves the growthshooting, partial pivoting0.0018shooting, τ = 0.10.56Wilkinson, τ = 0.10.1800.10.20.30.40.50.60.70.80.9110⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1one minus the thresholdstored noise that halves the growthshooting, T = 12, h = 0.3Wilkinson, n = 24the same growth at every thresholdheld by a choice noise must push further to reverse

A threshold that holds the growth still

Partial pivoting's large growth on the shooting matrix rests on a margin of 5.6·10⁻³, and Wilkinson's rests on exact ties, which a perturbation at rounding breaks. A sparse code pivots with a threshold instead, taking the sparsest row among candidates within a factor τ of the largest. At every threshold from 1 to 0.1 both matrices grow by exactly as much as under partial pivoting. But the growth is now held by row counts, which no perturbation of the values can reverse: at τ = 0.1 it survives noise on the stored entries up to 0.56 on the shooting matrix, three hundred times more, and 0.18 on Wilkinson's, where partial pivoting's goes at 10⁻¹⁶. The margin that predicts it is the chosen candidate's height above the threshold line, and it needs no division by the growth.

elimination · Growth step 0.3rate per interval1.3largest r that catches it1.3110¹10²10³10⁴trigger sensitivity rgrowth factor1.021.324dashed: e to the five-sixths of the stepthe trigger must be finer than the growth

A trigger finer than the growth

Threshold pivoting cannot remove the shooting matrix's growth of 1.1·10⁴, because at τ = 1 it is partial pivoting. The proposal was a factorisation that notices growth on a step and searches as rook pivoting does after it, paying for the stricter search only where growth is made. But no step makes much: the largest one-step rise is exactly the growing mode's rate over one shooting interval, 1.284 at a step of 0.3, and 1.133 at 0.15. A trigger finer than that rate holds the growth at 2.00 for about two thirds of rook's extra search; one coarser never fires. And a trigger fine enough searches as rook on a quarter of an ordinary random matrix's steps.

elimination · Growth

Named alongside it

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

Growth factorFill-inGaussian eliminationBackward errorMarkowitz costSparse LUSparsityPermutationSparse pivotingSymbolic factorisationBackward stabilityPartial pivoting

All concepts