Rational arithmetic — where it appears
Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.
An answer that is known
Almost every demonstration of numerical error estimates the error by computing the same thing more carefully. The Hilbert matrix does not need that: its inverse is a closed form in integers, so the true answer is available exactly and the error is measured rather than approximated.
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.
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.
The middle eigenvectors come back exact
The Sylvester–Kac matrix's eigenvectors are known exactly, and the middle ones were predicted to be lost before their eigenvalues, since the gap between neighbouring eigenvalues is only two. Started from the all-ones vector, inverse iteration seems to confirm it: half the middle eigenvectors are wholly wrong at order 80 while their eigenvalues keep five digits. But the all-ones vector has, in exact arithmetic, no component at all along every other eigenvector. From a random start the middle eigenvectors come back exact to fourteen digits at every order measured, while their eigenvalues lose up to eleven. What is lost is at the ends of the spectrum, in the smallest entries of eigenvectors whose entries span twenty decades — and never by more than three units of roundoff times that span.
The vector moves with its eigenvalue
The Sylvester–Kac matrix's middle eigenvectors came back exact to fourteen digits by inverse iteration while their eigenvalues lost eleven, and the route LAPACK uses — back substitution in the computed Schur form, then the accumulated orthogonal transformation — was predicted to lose them, at about the eigenvalue's error divided by the gap of two: 10⁻⁵ at order 80. It does, at 2.3·10⁻⁵, so the route is the whole of the difference: nine orders of magnitude between two ways of computing the same vectors from the same eigenvalues. The constant is not one over the gap. On all 274 eigenvectors whose eigenvalues are off by more than 10⁻¹¹, at every order from 40 to 80, the Schur route's error is 0.76 to 0.84 times its eigenvalue's error, and 0.79 is a number the exact vectors carry: how fast a unit Krawtchouk vector turns as its eigenvalue moves. The Schur route returns the right eigenvector of a wrong eigenvalue.
Named alongside it
The objects these essays reach for when they reach for this one.
Exact arithmeticBackward errorCertificateDeterminantEigenvalue condition numberExact ground truthFloating pointForward errorInteger relationInverse iterationLattice reductionNon-normality