Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Reduction order — where it appears

The order in which a parallel reduction adds its partial sums, decided by how many workers a runtime has and which of them finish first. Addition is not associative in floating point, so it is an input to the answer that no part of the program supplies.

Named by 14 essays across 2 fields — each of them below, with the objects they name alongside it.

110¹10⁻¹¹10⁻⁸10⁻⁵pieces the vector was divided intodistance from the exact sum, relativethe published boundκ · uone vector, one algorithmdistinct answers21runs26spread, in ulps2.3·10⁷κ of the sum10⁸bound ÷ worst error2.6·10⁴nobody chose pand no answer is the answer

The same program, twice

One vector of 4,096 numbers, one summation algorithm, one precision, twenty-six runs — and twenty-one different answers. Nothing in the program chose between them, every one of them satisfies the textbook bound, and the exactly rounded answer is not among them.

machine · reduction order 10²10⁴10⁶10⁸10¹⁰10¹²10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹κ of the sumrelative sizethe boundκ · umeasured spreadtwo curves and one constantspread ÷ κu, low0.25spread ÷ κu, high0.26bound ÷ spread, low7932decades swept10the spread is computablethe bound cannot see the order

A bound every answer satisfies

The classical bound on a summation error is correct, it covers all twenty-six answers one vector produced, and it is 7,932 times larger than the difference between them. A statement true of every ordering cannot say which ordering you got.

machine · reduction order 051210241536204810⁻¹⁷10⁻¹⁴terms consumederror accumulated so farone accumulator8 piecesthe steps are the partial sumspeak partial sum32the answer1.6·10⁻⁵error, one piece7.6·10⁻¹⁴error, 8 pieces2.4·10⁻¹⁴mean at p = 648.1·10⁻¹⁵the walk sets the sizeand nothing sets the value

Where the disagreement comes from

The error of a reduction is a walk whose step length is the spacing of the running total, not of the answer. That one sentence predicts the size of the disagreement to a factor of two, explains why dividing the work makes it smaller, and explains why the value cannot be predicted at all.

machine · reduction order κ of the suma component of b − Ax1.01·10¹⁷ad − bc, near-degenerate3.6·10¹⁶qᵢᵀqⱼ, an orthogonality check7.39·10¹⁵pᵀAp, a curvature409zᵀAz, a trace probe41.8rᵀr, a residual norm1measured, not assumedhighest10¹⁷lowest1above 10¹⁰3terms64sums of squares are safeand nobody decides anything from one

The vector that hides it

Every quick demonstration of a parallel sum uses positive numbers, and positive numbers are the one family where the effect is absent. Measured on six inner products this site already computes, the summation condition number runs from exactly 1 to 10¹⁷ — and the safe end is where nobody makes a decision.

machine · summation distinct answersone accumulator303eight pieces72compensated119pre-rounded1exact1worst error 1.17·10⁻⁸worst error 4.61·10⁻⁹worst error 4.96·10⁻¹⁰worst error 4.91·10⁻⁶worst error 0bitwise, or not at allpermutations400one accumulator303pre-rounded1its error4.9·10⁻⁶compensated error5·10⁻¹⁰accuracy and agreement are different propertiesand the accurate one is not the agreed one

The sum that cannot be wrong

Snap every addend to a common multiple before adding, and every partial sum is exact — so the order stops mattering, by construction rather than by luck. Four hundred permutations return one value where an ordinary reduction returns three hundred and three.

machine · reproducible summation 02468101210⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²binary64 operations per elementrelative errorruntime orderfixed partitionpre-rounded, one passpre-rounded, two passescompensated (Kahan)exact accumulationwhat determinism costsreproducible policies3cheapest, ops/element3its error0.0024two passes10⁻¹⁷ordinary, 8 pieces3.6·10⁻⁷nothing is cheap and exactthe middle of the figure is the answer

Accuracy and agreement are different properties

The most accurate policy on this site's summation figure returns 119 different answers, and the one that returns a single answer is four orders less accurate. Neither property implies the other, and the vocabulary has one word for both.

machine · reproducible summation 110¹670673676679682685688691694pieces the inner products were summed initerations to the tolerance674, the cheapest run690, the dearestthe same solve, pricedpartitionings run13distinct counts11spread, per cent2.4best forward error6.3·10⁻¹¹worst1.1·10⁻¹⁰one matrix, one toleranceand the cost is the machine's

A stopping test is a race

One matrix, one right-hand side, one tolerance, thirteen partition counts — and eleven different iteration counts between 674 and 690. Every run converged, every answer is right to the accuracy asked for, and what differs is the bill.

machine · stopping test -12-10-8-610⁻¹³10⁻¹¹10⁻⁹10⁻⁷log₁₀ of the residual tolerance asked forforward error of the answer1.34×1.48×1.71×1.17×the band does not closetolerances swept4runs at each7accuracy gained1.5·10⁶ratio at 10⁻⁶1.3ratio at 10⁻¹²1.2the bars falland they keep their height

The tolerance that buys no agreement

Ask for four more orders of accuracy and you get them — the answers improve by a factor of 1.5 million. The ratio between the best and the worst run is 1.34, 1.48, 1.71 and 1.17 across the same sweep. The band falls and it does not close.

machine · stopping test 13579111310⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹index ksingular valueσ₁ · 10⁻¹⁴, the thresholdone matrix, three rankspartitionings7lowest rank10highest rank12threshold7.3·10⁻¹⁴σ₁7.3the curves separate in the noiseand the threshold is drawn through it

A rank that depends on the thread count

One 60 × 14 matrix, one threshold, seven partitionings of the inner products that build its Gram matrix — and numerical ranks of 12, 12, 12, 10, 10, 11 and 11. Not a digit of an answer: the number of columns a model built from this matrix would have.

machine · rank 5860626466687010⁻¹⁷10⁻¹⁶terms in the dot productmean relative errorthe kernel changes herea constant in a libraryone accumulator3.2·10⁻¹⁷four accumulators2.1·10⁻¹⁷step at the cutoff1.6cutoff64the problem did not changethe loop did

The length that changes the kernel

A dot product's accuracy steps by a factor of 1.57 between 63 and 64 terms, on vectors drawn identically at both lengths. Nothing about the problem changes there. A library switches from one accumulator to four, at a constant in somebody else's source file.

machine · algorithm selection -15-13-11-910⁻¹³10⁻¹¹10⁻⁹10⁻⁷log₁₀ of the defect, relative to the entry it sits inhow far the answer movedthe machine, on its owna tolerance with two sidesmachine band3.2·10⁻¹²smallest caught10⁻¹²its nearest run4.6·10⁻¹²defects hidden3window, factor1.4below the band nothing is visibleand above it everything is

What a regression test can ask for

The machine's own variation on one solve is 3.2·10⁻¹², and the smallest defect whose answers clear it is one part in 10¹². The tolerance exists, it is bracketed on both sides by a factor of 1.42, and it is neither zero nor the 10⁻⁸ that usually gets typed.

machine · regression tolerance the problem you posedA = H12b = A·(1, 2, …, 12)the problem it answered exactlyA + δA, b + δb‖δ‖ / ‖A‖ = 1.8·10⁻¹⁷the answer you wantedx = (1, 2, …, 12), exactlythe answer you gotx̂, wrong by 0.02 relativebackward error 1.8·10⁻¹⁷forward error 0.02κ = 1.8·10¹⁶κ · η = 0.33, and the measured forward error is 0.02.The algorithm is not at fault. The problem is.H12, LU with partial pivotingresidual and error differ

The fifth author

Four authors of a wrong answer have been named on this site and each is a statement about one computation. The fifth is not: it is what separates two computations that are both correct, it is a backward error of measurable size, and no residual, bound or condition number contains it.

error · backward error 110¹-101pieces the inner product was summed inqᵢᵀqⱼ ÷ 2.78·10⁻¹⁷the true valuethe norm and its partsdistinct values5of runs10pairs with no fixed sign125of pairs1128κ of this inner product1.1·10¹⁷‖QᵀQ − I‖ moves by1the aggregate is stableand no entry of it is

An inner product with no fixed sign

‖QᵀQ − I‖ is how this site turns "orthogonal" into a number, and across ten partitionings it moves by 2.4%. The entries it is built from are not so lucky: 125 of the 1,128 off-diagonal pairs take both signs, and one of them takes five different values including zero.

machine · orthogonality 00.10.20.30.40.50.60.70.80.9110⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹decay rate of the spectrumrelative spread of the rank-k errorthe seedthe machinetwo kinds of variationseeds drawn8partitionings7seed spread0.31machine spread9.8·10⁻¹⁶ratio3.1·10¹⁴one of these is recordedand it is the large one

The variation that comes with a seed

A randomised low-rank approximation's error moves by 31% between draws and by 10⁻¹⁵ between partitionings of one draw. In the one field on this site whose answer already comes as a band, the machine is inside the width of the line — and it is still there.

machine · run to run variation

Named alongside it

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

Run-to-run variationBitwise reproducibilitySummation condition numberUnit roundoffError accumulationResidualToleranceAlgorithm selectionBackward errorConjugate gradientsExact accumulationGram matrix

All concepts