Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Rounding — where it appears

Replacing an exact result by the nearest representable number. Every claim about accuracy here is a claim about the accumulation of these, and which direction they lean is set by the mode rather than by the precision.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

4 numbers, coefficients to 30found relations72with a gap of a digit or more54accidents56accidents with that gap05101520253001234decimal digits the numbers are scaled togap, in digitsexact relation foundnot a relationthe room a relation hashorizontal: a gap of one digitdashed vertical: the digits a double has

The room a relation has to stand out

A lattice search for an integer relation returns its shortest vector, and the proposal was to return the gap to the next one as well, so a caller could tell a relation from an accident. Measured, the gap is a certificate with a budget: the digits the numbers really have, shared among all but one of them, less the size of the relation. A found relation's gap sits half a digit under that budget, accidents stay near zero, and a one-digit gap vouches for 81 of 96 relations among three numbers and for 1 of 35 among six. The test numbers the proposal came from turned out to have relations of their own.

exact · Lattice reduction 110¹10²10³110¹10²10³the accumulating quantity, relative to its first valuethe error, relative to its first valuethe bounds: slope 1what all three do: slope ½three mechanisms, one exponenta left-to-right sum0.49a chain of rotations0.55a residual recurrence0.51every bound's slope1spread of the three0.067a bound is a sum of the roundingsand the roundings have signs

Three walks and one bound

A left-to-right sum, a chain of three thousand rotations and a conjugate gradient residual recurrence share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against. All three come out at a half — 0.486, 0.554 and 0.507 — and nothing is rescaled.

arithmetic · Summation 6 numbers, coefficients to 30relations, both agree34relations, they differ1accidents, both agree5accidents, they differ88510152025300123decimal digits the numbers are scaled togap, in digitsrelation, both agreerelation, they differaccident, both agreeaccident, they differhorizontal: a gap of one digitdashed vertical: the digits a double has

Two precisions guard the other edge

Run a lattice relation search at N digits and again at N/10, and accept its answer only if both runs return the same vector. Among six measured numbers, where the gap between the shortest and next vector vouches for one found relation in 35, the two runs agree on 34. Past a double's sixteen digits they never once agree on an accident, 0 of 561. They do agree on 62 accidents at fifteen digits or fewer — approximate relations that really are the shortest vector there — and the gap, which cannot see a relation among six numbers, can see those.

exact · Lattice reduction 10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁴10⁻¹10¹²10²⁵10³⁸10⁵¹10⁶⁴δ, the gap between consecutive eigenvaluesrelative error in eᴬan answer with no correct digitsV f(Λ) V⁻¹scaling and squaring‖A(δ) − A₀‖exact eigenvalues throughoutκ(V) at the smallest δ3.3·10⁸²eigen route2.9·10⁶⁵scaling and squaring4.1·10⁻¹²distance to the limit7·10⁻¹²the eigenvalues are the diagonaland they are exact at every stop

The series that has to be squared back

The Taylor series for the matrix exponential is not wrong — every term is computed correctly — and on Moler and Van Loan's two-by-two its largest term is 5.4 million times the answer it sums to. The method that replaces it scales the matrix down and squares the result back, and both halves of that sentence cost: too few squarings and the approximant is out of range, too many and each one doubles the rounding.

spectra · Matrix function pooled over every settingaccidents, one digit apart62accidents, three digits apart5relations lost on the way109110¹10²10³digits between the two searchesruns123relations keptaccidents acceptedlogarithmic vertical axiseach digit apart divides the accidents by three or four

The digits between the two searches

A lattice relation search run at N digits and again at N/10 never agrees with itself on a relation among the rounding, but it does agree on 62 approximate relations at fifteen digits or fewer — accidents that really are the shortest vector there, and that dropping a digit was said to be unable to dislodge, since a combination that cancels to D digits cancels to D − 1. It cancels, but it stops being the shortest. Two digits apart the accepted accidents fall to 17 and three digits apart to 5, while the relations kept fall from 711 of 755 to 652 and 602 — every one of the losses at fifteen digits or fewer. Past a double's sixteen digits the separation costs nothing and buys nothing.

exact · Lattice reduction

Named alongside it

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

CertificateExact arithmeticFloating-pointLattice reductionBackward errorDeterminantExact ground truthInteger relationOrthogonality defectRational arithmeticCancellationCompensated summation

All concepts