Numerical Linear Algebra
where the algebra stops being the arithmetic
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The thread: Where the algebra stops being the arithmetic

Every identity in linear algebra is exact, and every implementation of it is not. The gap is not a defect to be apologised for — it is the subject. QᵀQ is the identity in the algebra and is measurably not the identity in the arithmetic, and the difference between those two sentences is what this site is about.
A = H8 · κ = 1.5·10¹⁰ · both factorisations reconstruct A to 6·10⁻¹⁷the diagonal is 1 in both — every column is a unit vector either way1.000000000001.000000000001.000000000001.000000000001.00000.002-0.002000001.0000.125-0.13300000.0020.1251.000-1.0000000-0.002-0.133-1.0001.000classical Gram–Schmidt1.000000000001.000000000001.000000000001.000000000001.000000000001.000000000001.000000000001.000Householderclassical ‖QᵀQ − I‖1.4Householder ‖QᵀQ − I‖1.4·10⁻¹⁵largest off-diagonal 1 against 3.1·10⁻¹⁶length is not angle Orthogonality, measured

Orthogonal is a number

"Q is orthogonal" is a claim about a measurable quantity, ‖QᵀQ − I‖, and on the eight-by-eight Hilbert matrix two standard algorithms return 10⁻¹⁵ and 1 for it. The one that returns 1 still reconstructs the matrix perfectly, which is why nothing warns you.

 the problem you posedA = H10b = A·(1, 2, …, 10)the problem it answered exactlyA + δA, b + δb‖δ‖ / ‖A‖ = 2.3·10⁻¹⁷the answer you wantedx = (1, 2, …, 10), exactlythe answer you gotx̂, wrong by 2.7·10⁻⁴ relativebackward error 2.3·10⁻¹⁷forward error 2.7·10⁻⁴κ = 1.6·10¹³κ · η = 3.6·10⁻⁴, and the measured forward error is 2.7·10⁻⁴.The algorithm is not at fault. The problem is.H10, LU with partial pivotingresidual and error differ Two errors, and whose fault they are

The exact answer to a nearby problem

A good algorithm does not give an approximate answer to your problem. It gives the exact answer to a problem very close to yours — and once that is the definition, a wrong result has two possible authors and they can be measured apart.

 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 Sparsity, and what elimination costs

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.

 nothing8.3%the right-hand side16.5%the matrix, slowly85.9%everything100.0%what can be reused: the answerwhat can be reused: the factorisationwhat can be reused: the factorisation, for a whilewhat can be reused: nothingshare of the cost of a sequence that shares nothingwhat 12 members cost in factorisationssame: factorisations1rhs: factorisations1drift: factorisations4independent: factorisations12what changes between the membersdecides what may be carried When the problem arrives again

The problem that arrives again

A hundred and thirty essays have solved a system once and measured how wrong the answer was. Almost no computation is shaped like that. A solve is one step of an outer loop, its answer is an input rather than a deliverable, and four quantities treated here as accuracy requirements turn out to be assets with a shelf life.

 everything Ax can reachb = (1.1, 0.4, 1.5)Ax, the closest reachable pointr = b − Ax‖Aᵀr‖ / (‖A‖‖r‖)1.7·10⁻¹⁶‖b‖² − ‖Ax‖² − ‖r‖²1.3·10⁻¹⁵‖r‖1.3200 random nearby points of the plane were tried; none is closer.a 3×2 system, Householder QRperpendicularity is checked Least squares, and the road not to take

The projection and the right angle

The least-squares solution is the one whose residual is perpendicular to everything the columns can reach. That is not a mnemonic — it is an equation, Aᵀr = 0, and the computed answer satisfies it to 10⁻¹⁶.

 10²10³10⁴10⁵matrix size ncountoperations, bothwords, unblockedwords, blocked (b = 6)the answer does not move‖PA − LU‖/‖A‖, unblocked2.8·10⁻¹⁶‖PA − LU‖/‖A‖, blocked2.8·10⁻¹⁶difference between them0the dashed curve is both orderings' operation countthe solid pair is what they cost Where the flop count stopped predicting the time

The same arithmetic at a different price

A blocked and an unblocked elimination perform 72,568 operations each — the same operations, associated differently — choose the same pivots, and return a factorisation identical to the last bit: ‖PA − LU‖/‖A‖ = 4.487946226420872·10⁻¹⁶ in both. One of them moves 41,332 words between fast and slow memory and the other moves 19,476.

 -1-0.582271-0.1645420.2531860.6709151.088641.506370eigenvalue4 negative10 positivecounted before it was formedpositive10negative4at zero0innermost ratio39the zero block is a theoremand so is the count either side of it The matrix a constraint makes

The zero that is not a missing entry

A constrained minimisation produces a matrix with a zero block, and the zero is a theorem rather than a sparsity pattern. No pivot order makes it positive definite, no precision changes that, and Cholesky does not fail somewhere on it — it fails at the first constraint row, on a number the problem already contained.

 00.0939569-2-1.24498-0.4899610.2650581.020081.7751real partimaginary parttwo routes, one spectrumeigenvalues16rows8complex16against the closed form2.1·10⁻¹⁵n rows and 2n eigenvaluesso the eigenvectors are not a basis The eigenvalue problem that is not linear

A matrix that depends on its own eigenvalue

A damped structure does not produce Ax = λx. It produces (λ²M + λC + K)x = 0, where the matrix whose null vector is wanted is a function of the number being solved for — so there is nothing to factorise, an n × n problem has 2n answers, and the eigenvectors cannot be a basis.

 10⁻¹110¹10²10³10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹frequency ω|H(iω)|one function, two routesstates24relative, ω ≤ 10010⁻¹⁴against the terms3.9·10⁻¹⁵‖H‖∞0.0924 statesone function of one variable Reduction, and what a model is for

A model that is a rational function

A state matrix has a hundred thousand rows and the thing anyone wants from it is a function of one complex variable. The number that says how much of that size was ever the complexity is a rank — and the rank a derivation writes down cannot be computed, while one built from samples alone can.

 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 answer that depends on the machine

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.

 21-13-3-121-212-443-12as givenrows in the order 1 2 3 443-1201.251.252.502.51.5-30-0.5-0.52after step 1pivot 443-1202.51.5-3000.5400-0.21.4after step 2pivot 2.543-1202.51.5-3000.540003after step 3pivot 0.5‖PA − LU‖/‖A‖0largest multiplier0.75row order 4 3 2 1the pivot is chosen Elimination, and the swap

Elimination is a sequence of choices

Gaussian elimination is taught as a procedure with no decisions in it. There is one decision at every step — which row to use — and every stability property the algorithm has comes from making it well.

 [½, 1)[1, 2)[2, 4)0.5124gap 0.125gap 0.25 — twice as wide8 values per octavespacing doubles at each power of two The arithmetic underneath

What a float can hold

The representable numbers are not a fine fuzz spread evenly over the line. They are evenly spaced inside each power-of-two interval and twice as far apart in the next one up, and almost everything else in this subject is a consequence of that one fact.

 the matrix, measuredvertices40edges223‖L·1‖∞0zero eigenvalues1components, by search1λ₂1.5laid out at its own eigenvectorsand the row sums are exactly zero The matrix that is a graph

A matrix with no numbers in it

A graph arrives as vertices and edges. Two different matrices can be built from it, they answer different questions, and one of them has a null vector that is exact — the only object on this site whose kernel is known before anything runs.

 3456789101112110¹10²10³10⁴10⁵10⁶nwidest intermediate, in bitsrationals, not reducedfraction-free · rationals reduced · the answerone answer, three widthsn12answer40Hadamard bound52fraction-free40reduced rationals39unreduced1.4·10⁶the error is zero on every curvethe cost is the length of the numbers Exact arithmetic, and what it costs instead

An answer with no error in it

An integer matrix eliminated over the rationals rounds nothing, so the forward error is zero, the residual is the zero vector, and the identity this site is built on has no terms left. The cost does not vanish with the error. It moves into the length of the numbers, where three correct routes differ by four orders of magnitude.

 051015202530354010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, in orderσ ⁄ σ₁eight digitsan independent draw per entry1 ⁄ ra cliff, and a control1/r rank at 10⁻⁸5log r rank at 10⁻⁸5noise rank at 10⁻⁸96σ₂ ⁄ σ₁0.024σ₆ ⁄ σ₁2.8·10⁻⁹the block has full rankand five useful columns Neither sparse nor dense

A block nobody can call sparse

A 96 × 96 block of a kernel matrix has ninety-six nonzero singular values and five that matter. It has no zero entries, it is not described by fewer numbers than it contains, and neither of the two ways this collection already knows to make a large matrix affordable applies to it.

 03672108144180216024681012eigenvalues in orderλthe closed formmarks: the assembled matrix, decomposeda spectrum nobody computedrows of the matrix216numbers that describe it108λ smallest0.59λ largest11worst |computed − exact|7.1·10⁻¹³the matrix is never neededand neither is its decomposition When the index is a tuple

An index that is a pair

A discretisation on a two-dimensional grid of n points a side has n² unknowns and a matrix with n⁴ entries — 10⁸ at n = 100. What that matrix is instead is two Kronecker products of an n × n matrix, which is 2n² numbers, and nothing has been approximated: assembling it was the mistake.

 10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³110⁻⁴10⁻³10⁻²10⁻¹1inner tolerance η, relative residual of the linear solvedistance from the root after the stepd² = 0.00138distance before the step, 0.03721093547231126the number by each point is the iterations it costeleven decades, one landing placedistance before the step0.037its square0.0014where η = 10⁻³ lands0.0025where η = 10⁻¹⁴ lands0.0025iterations for the first354iterations for the second1126the accuracy that is thrown awaymeasured against a root that is known When the problem arrives again

The accuracy that is thrown away

A Newton step is the exact answer to a linearised problem, and the linearisation is wrong at second order. So there is a floor under how close the step can land, the floor is the square of where it started, and eleven decades of inner tolerance below it buy the same four digits at four times the price.

 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 Sparsity, and what elimination costs

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.

 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 The answer that depends on the machine

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.

 10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²110⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹xrelative error of the computed value(1 − cos x)/x², as written2 sin²(x/2)/x²no digits left at allbinary64 throughoutone function, two spellings · zero below 1.5·10⁻⁸ The arithmetic underneath

Cancellation takes the answer, not a digit

Subtracting two nearly equal numbers is exact. That is what makes it dangerous — the subtraction introduces no error at all, it exposes error the operands were already carrying, and the exposure can consume every significant figure at once.

 10²10².³10².⁵⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁶10⁻¹10⁻⁰.⁵1rows in the sketchworst relative distortiondimension 64dimension 2565 seeds per point, band is best to worstthe dimension does not appear Randomised, and the guarantee that changes kind

The dimension does not appear

A random projection preserves the lengths of a set of vectors to within a distortion that depends on how many vectors there are and not on how many coordinates each one has. That is the fact the whole field rests on, and it is genuinely surprising.

 for each previous column i, subtract the projection of column j onto qᵢclassicalr[i][j] = qᵢ · a[j] ↑ the ORIGINAL columnv = v − r[i][j] · qᵢthe three worst |qᵢ · qⱼ|:columns 7 and 8: 1columns 6 and 8: 0.13columns 6 and 7: 0.13modifiedr[i][j] = qᵢ · v ↑ what is LEFT of itv = v − r[i][j] · qᵢthe three worst |qᵢ · qⱼ|:columns 1 and 8: 4.4·10⁻⁷columns 2 and 8: 2.7·10⁻⁷columns 3 and 8: 2.4·10⁻⁸The two R factors agree to 1.2·10⁻⁶ relative. The two Q factors do not.the 8×8 Hilbert matrixone word, eight orders Orthogonality, measured

Two Gram–Schmidts

One argument changes. Classical Gram–Schmidt projects the original column onto each previous direction; modified projects what is left of it. In exact arithmetic the coefficients are identical. In floating point they differ by eight orders of magnitude in the thing that matters.

 10⁻³10⁻²10⁻¹110¹12345678910conductance, and the two bounds on itpathcyclegridbarbelltwo blockshypercubepreferentialstarcompletethe bar is the inequalitythe dot is the graph The matrix that is a graph

Two Laplacians of one graph

The combinatorial Laplacian D − A and the normalised one, which conjugates it by the inverse square roots of the degrees, are built from the same object, are not similar to each other, and answer different questions. On a graph whose degrees are equal they coincide. On one whose degrees span an order of magnitude their second eigenvalues are sixteen times apart.

 0246810⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹log₁₀ γ, the change of unitsrelative errorforward errorη, the quadraticη, the linearisationagainst a closed formη(linearisation), worst7.6·10⁻¹³η(quadratic), worst1.2·10⁻⁴forward error, worst0.0013coefficient spread4.2·10¹⁵the solver is right at every stopabout a problem nobody asked The eigenvalue problem that is not linear

A backward-stable answer to a problem nobody asked

One quadratic eigenvalue problem, in nine systems of units, with a change of variable that is exact in both directions. The residual the solver prints stays at the rounding level at every stop. The answer loses eleven orders of magnitude, and the two facts are consistent.

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