Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Fill-in — where it appears

An entry that is zero in a matrix and nonzero in its factor. Eliminating a variable couples everything it touched, so the number of them is decided by the elimination order, and it is what makes a sparse factorisation expensive rather than the arithmetic.

Named by 19 essays across 4 fields — each of them below, with the objects they name alongside it.

the matrix, lower triangle408 entriesits Cholesky factor1739 entries · 1331 created→‖A − LLᵀ‖/‖A‖1.4·10⁻¹⁶fill, symbolic1331fill, numeric1331n = 144 · density 3.2% · bandwidth 12same matrix, renumberedthe answer is identical to rounding

The factor is not sparse

A sparse matrix has a factor that is not sparse, and the gap between them is the entire reason iterative methods exist. The entries elimination creates can be counted before any arithmetic runs, from the graph alone.

sparsity · Fill natural1739reverse Cuthill–McKee1354minimum degree1026nested dissection1413matrix: 408 entries · dense factor: 10440bandwidth 12 · 4.26× the matrixbandwidth 12 · 3.32× the matrixbandwidth 123 · 2.51× the matrixbandwidth 108 · 3.46× the matrixn = 144, five-point stencilevery ordering fills in; none avoids it

The order decides the memory

Four elimination orderings on one matrix give factors of 1,739, 1,354, 1,413 and 1,026 entries. All four factorisations are exact, all four return the same answer, and the one with the better asymptotics is not the one that wins.

sparsity · Ordering the matrix43 entriestip eliminated first253 entriestip eliminated last43 entries‖A − LLᵀ‖/‖A‖, tip first1.4·10⁻¹⁶‖A − LLᵀ‖/‖A‖, tip last0dense factor is n(n+1)/2 = 253 · sparse factor is 2n − 1 = 43one row swapped to the endnothing numerical chose between them

Two ends of the same arrow

One matrix, one row moved from the front of the elimination order to the back, and the factor goes from completely dense to no fill at all. Both factorisations are exact to rounding, and nothing numerical chose between them.

sparsity · Fill 051015202530354010⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹iteration‖r‖ / ‖b‖plain CGIC(0) CGwhat the preconditioner didκ(A)48κ(L⁻¹AL⁻ᵀ)5.1‖A − LLᵀ‖/‖A‖0.0832D Laplacian, n = 100√κ ratio predicts 3.07×

Changing the condition number on purpose

Preconditioning is usually introduced as a trick that makes an iteration converge faster. It is not a trick. It is solving a different system with the same solution and a condition number chosen rather than inherited, and the new condition number is computable.

iterative · Preconditioning 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 the bound211no pivoting127τ = 0.00198τ = 0.003100τ = 0.01106τ = 0.03111τ = 0.1116τ = 0.3117τ = 1138entries in Ua bound, and its slackthe bound, from the graph alone211the worst that occurs138loose by1.5no arithmetic was done to compute the bound — only the pattern of AᵀA and its elimination graphGeorge and Ng: U fits inside chol(AᵀA), whatever the swapsprovable, cheap, and loose

What the symbolic phase can only bound

Without pivoting, the fill can be computed from the graph and the count is exact — 233 predicted, 233 measured. With pivoting it is 233 predicted and 242 measured, and what survives is a bound that is right at every threshold and loose by 1.7 times at the largest grid drawn.

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 0481216202402468101214unknowns on the separatorcolumns above 10⁻⁸the same matrix, renumberedin the separator's own orderthe ordering the geometry hands overseparator 73separator 236renumbered, largest11the block, largest11share of the square stored0.52the fill is totaland it is not independent

The fill that is not independent

Eliminate both halves of a grid and what is left on the separator is 100 per cent nonzero — the sparsity field's result, unchanged. Its off-diagonal block is 11 by 12 and six columns describe it to eight digits. Renumber the separator and the same block needs all eleven.

sparsity · Fill natural113 predicted · 113 countedminimum-degree63 predicted · 63 countedreverse Cuthill–McKee63 predicted · 63 countedthe shaded entries are fill: zeros of K that the factorisation makes nonzeroallocated before the numbersnatural113minimum-degree63reverse-cuthill-mckee63predicted minus counted0the symbolic phase decides the memoryand nothing later is allowed to argue

An ordering that does not wait for the numbers

A sparse factorisation's memory is decided by an ordering computed from the graph, and its stability by pivots computed from the values, and the two decisions fight. On one family of matrices they do not — the ordering can be chosen for fill alone, and the fill the symbolic phase predicts is the fill the factorisation produces — exactly, not as a bound.

sparsity · Sparse pivoting A6×6B6×6I ⊗ A + Bᵀ ⊗ I36×36one equation, two objectsentries in A and B72entries in the coefficient matrix1296two routes, relative gap1.7·10⁻¹⁶‖AX + XB − C‖/‖C‖1.4·10⁻¹⁶the small squares are the problemand the large one is the notation

The elimination the matrix does not need

The Kronecker form of AX + XB = C is dismissed with a hundred million entries and (2/3)n⁶ operations. Both price an elimination, and after the reduction both routes take, the matrix has exactly n³ nonzeros, none of them above the block diagonal, and nothing left to eliminate.

structure · Matrix equation least possible76minimum degree76worst of 300 tie-breaks81reverse Cuthill–McKee85natural order9920 unknownstie-breaks at the least0.44minimum degree ÷ least1every order searched, by the set already eliminatedthe greedy rule found the minimum

The least fill there is

Finding the elimination order with the least fill is NP-hard, and that is a statement about the hardest graph and the largest size. On a graph of twenty vertices every one of the 20! orders can be searched at once, through the million sets of vertices already eliminated, and the least fill is a number. On the 4×4, 4×5 and 3×7 grids minimum degree finds it exactly. On eighty random sparse graphs of eighteen vertices it finds it on 53 and misses by at most 7.6 per cent, and on every one of the eighty some breaking of its ties finds it.

sparsity · Ordering all 2^16 subsets searchedboth minima, one order0.97minimum degree optimal0.82worst fill overshoot1.1worst work overshoot1.111.021.041.0611.041.081.121.16fill ÷ the least filloperations ÷ the leastovershoot squaredon the diagonal: the two overshoots are equalon the upper curve: the arithmetic overshoot is the square

Two minima that are one minimum

The order that decides the memory found the operation count behaving like the square of the fill, which leaves room for an order with slightly more fill but a shorter heaviest column to do less arithmetic. Searched exactly over every elimination order on forty graphs, that order does not exist: one order attains both minima on thirty-nine of forty, and on the fortieth the least-fill order's arithmetic is 1.0099 times the least. Minimum degree attains both on the same thirty-three graphs and neither on the same seven.

sparsity · Ordering 01234567110¹10²levelstored entries per rowalgebraicgeometrictwo complexities, one hierarchygrid complexity3.1operator complexity18geometric, for comparison1.5a ring plus random chords — nothing is anywherelevel 6 is 100% dense

A hierarchy with no grid behind it

On a graph Laplacian the algebraic V-cycle converges at 0.199 a cycle, its grid complexity is an unremarkable 3.05, and its operator complexity is 17.7 — one level of forty-one unknowns is entirely dense. The number people quote is the one that does not measure the work.

iterative · Algebraic multigrid total operationswork, no dissection10⁵work, full dissection1.5·10⁵work, one level1.3·10⁵on unbounded processorscritical path, no dissection4.1·10⁴least, at depth 43.4·10⁴critical path, one level5.4·10⁴10⁵switching depthoperations01234568total operationscritical pathdepth 0 is minimum degree, depth 8 is nested dissectiondepth 1 is worse than both

The depth that is worse than both ends

Nested dissection to a chosen depth and minimum degree below it is the ordering codes ship, and sweeping the depth was supposed to find a setting that keeps most of dissection's parallelism for most of minimum degree's work. It does not exist: total work rises with the depth at every grid size, and one depth — the first — is worse than both extremes on work and on the critical path at all four sizes measured. One bisection buys nothing because there is no recursion under it to amortise the separator.

sparsity · Ordering 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 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 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

Named alongside it

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

SparsitySymbolic factorisationFill-reducing orderingGaussian eliminationPermutationGrowth factorThreshold pivotingMinimum degreeElimination graphNested dissectionSparse pivotingBackward error

All concepts