Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Floating point — where it appears

The representation of real numbers as a sign, an exponent and a fixed number of significand bits. Its spacing doubles at every power of two, so it holds every integer up to 2^53 exactly and not one consecutive pair above it.

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

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

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.

arithmetic · Floating point 57911131517192110⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹vertices of the complete graphrelative sizethe algorithm is not the problemsizes counted18worst relative error9.4·10⁻¹⁵first wrong at n =16digits in the answer17spacing of doubles there8largest integer a double holds9·10¹⁵the elimination stays at the rounding levelthe format runs out

A count that comes out of a determinant

The number of spanning trees of a graph is the determinant of its grounded Laplacian, so it is a whole number known in advance. The elimination that computes it is backward stable at every size — and from sixteen vertices the answer is wrong, because the count has seventeen digits and a binary64 has sixteen.

graph · Graph elimination 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 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 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.

CertificateDeterminantExact arithmeticLattice reductionRoundingBackward errorInteger relationOrthogonality defectRational arithmeticCospectral graphsCounterexampleError accumulation

All concepts