Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Woodbury identity — where it appears

The formula for the inverse of a matrix plus a low-rank correction, in terms of the inverse of the matrix and a small dense solve. It is what turns a hierarchical solve into two solves one level down, and it assembles nothing.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

567891010⁴10⁵10⁶10⁷10⁸10⁹log₂ nmultiplicationsdense factorisation, n³⁄3the recursion, countedwhere the format starts payingratio at n = 641.5ratio at n = 5120.16exponent, first doubling2.1exponent, last doubling1.7backward error1.4·10⁻¹⁰cheaper is a sizenot a property

Where the format starts paying

A hierarchical solve costs 1.48 times a dense factorisation at 64 unknowns and 0.16 times it at 512. The crossover is between 64 and 128, it walks right when the accuracy is tightened, and the exponent between consecutive sizes is 2.13, 1.93, 1.74 — falling towards one and never arriving.

cost · Hierarchical solve -9-7-5-3-1110⁵10⁶10⁷log₁₀ ε of the preconditionermultiplications for the whole solveno preconditioner: 41 stepsiteration count, rescaledmultiplications spenttwo curves, opposite waysκ21steps, no preconditioner41cheapest ε0.5its rank1tightest ⁄ cheapest2.4the count is what is printedand the cost is what is spent

The accuracy worth paying for

Used as a preconditioner, a hierarchical representation gets better at every accuracy — the iteration count falls monotonically all the way to the tightest tolerance. The total work does not. Its minimum sits at a rank-one preconditioner on an easy problem and six decades further along on a hard one.

hierarchy · Hierarchical solve 10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³relative compression error, the representation‖b − Ax‖ ⁄ ‖A‖‖x‖, the solveequala backward error, chosenslope0.99representation at 10⁻⁸1.2·10⁻⁹backward error there1.3·10⁻¹⁰representation at 10⁻¹²1.2·10⁻¹³backward error there2.5·10⁻¹⁴the compression is not an approximationit is a perturbation of the problem

An accuracy that is a backward error

Every backward error on this site is something an algorithm produced and somebody then measured. This one is a line in the program. Solving with a compressed matrix gives a residual that is the compression's own error, at a slope of 1.000 over ten decades, so the knob that sets the storage sets the backward error directly.

error · Backward error 10⁻¹⁷10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10¹10³10⁵σ — how far the periodic wrap is from singularrelative forward error10⁻¹10⁻²10⁻³10⁻⁴10⁻⁵10⁻⁶10⁻⁷10⁻⁸10⁻⁹10⁻¹⁰10⁻¹¹10⁻¹²periodic correctionκ(C)·uelimination on Tn = 64, median of five answersperiodic correction, σ = 10⁻⁸1.2·10⁻⁴elimination on T, σ = 10⁻⁸1.1·10⁻¹⁴κ(C) at σ = 10⁻⁸4·10⁸κ(T) at σ = 10⁻⁸1712the same matrix T in every columnonly the wrap it is solved through changes

The circulant the problem did not contain

A matrix that differs from a circulant in two corner entries can be solved through the circulant, by a transform and a two-by-two correction, and the cost claim is exact. The accuracy claim is not. On tridiag(−1, 2 + σ, −1), whose condition number stops at 1,712, the correction is wrong by 1.2·10⁻⁴ at σ = 10⁻⁸ while elimination is right to 1.1·10⁻¹⁴ — because the periodic neighbour is singular at σ = 0 and the two-by-two system inherits that. Solved by Cramer's rule, as here, the two amplifications multiply; a later measurement found that a pivoted solve of the same two-by-two system removes the second.

structure · Circulant -7-6-5-4-3-210⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²110²10⁴log₁₀ of the twist's distance from landing, in turnsrelative forward errorsimple zero, landed on at 0simple zero, landed on at 0.3double zero, d = 2n = 64, twist 10⁻⁵ of a turn from landingpair: error, κ(A) = 4.1·10⁶5.4·10⁻¹²single: error, κ(A) = 3.8·10⁶3·10⁻⁸double: error, κ(A) = 4.2·10¹²7515the same condition number of the wrapa different number of modes near zero

Two near-zeros cost less than one

Solve a well-conditioned tridiagonal matrix through a nearly singular wrap and the correction's accuracy is not set by how singular the wrap is. At κ = 4·10⁷ one wrap returns the answer to 8.8·10⁻¹¹ — better than κ·u — and another, at κ = 3.8·10⁷, returns it to 3.3·10⁻⁶. The difference is how many of its samples sit near the symbol's zeros. A real wrap lands on a conjugate pair, a rank-two correction absorbs the pair exactly, and its two-by-two system has condition number 1.00 — which mattered because that system was solved by Cramer's rule; solved with pivoting, the single landing costs what the pair costs.

structure · Circulant size n163264128(L + s)²1 stepκu 1.2·10⁻¹²1 stepκu 1.9·10⁻¹¹1 stepκu 3.1·10⁻¹⁰1 stepκu 4.9·10⁻⁹(L + s)³1 stepκu 1.2·10⁻¹⁰1 stepκu 7.9·10⁻⁹1 stepκu 5.1·10⁻⁷1 stepκu 3.2·10⁻⁵(L + s)⁴1 stepκu 1.3·10⁻⁸1 stepκu 3.3·10⁻⁶2 stepsκu 8.4·10⁻⁴no first solveκu 2.2·10⁻¹(L + s)⁵3 stepsκu 1.3·10⁻⁶3 stepsκu 1.4·10⁻³no first solveκu 1.4·10⁰no first solveκu 1.4·10³(L + s)⁶3 stepsκu 1.3·10⁻⁴no first solveκu 5.6·10⁻¹no first solveκu 2.3·10³no first solveκu 9.5·10⁶κu: the wrap's condition number times the unit roundoffsteps are not a function of κu

Where one step stops being enough

One step of iterative refinement took the squared Laplacian's circulant-wrap solve to elimination's accuracy at every size, and the account was that each step multiplies the error by the wrap's condition number times the rounding, so one step suffices while that product is small. Raised to higher powers, the band tests the account and half of it holds: each step does contract by about κ(wrap)·u. The other half fails. The fifth power at sixteen points needs three steps with κ(wrap)·u near 10⁻⁶, where the third power at 128 points needs one with thirty times more, because the first solve starts up to a thousand times further from the answer than κ(wrap)·u says.

structure · Circulant

Named alongside it

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

Condition numberBackward errorCapacitance matrixCirculant matrixForward errorHierarchical matrixSymbolCancellationDiscrete laplacianFlop countIterative refinementNear-null space

All concepts