Francis's double shift — where it appears
Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.
Two shifts that are never formed
The double shift is defined as a factorisation of (A − μI)(A − μ̄I), which nobody computes. What is computed is the first column of that product — three numbers — and the bulge those three numbers create, pushed down the subdiagonal by n − 2 reflectors until it falls off the bottom.
Balanced is not symmetric
The Sylvester–Kac matrix is made of small integers, so a double holds it exactly, and its eigenvalues are the integers from −(n − 1) to n − 1 in steps of two. Every digit an eigensolver loses on it is therefore the solver's own, and it loses them at exactly the rate first-order perturbation theory predicts: the median error is half the prediction across 1,568 eigenvalues. Balancing, the preprocessing libraries apply for this kind of matrix, divides every condition number by about sixty and leaves their growth untouched, and at order 112 the unbalanced solver returns eighteen complex eigenvalues for a spectrum of integers.
Named alongside it
The objects these essays reach for when they reach for this one.
Bulge chasingDeflationEigenvalue condition numberExact ground truthHessenberg formHouseholder reflectionThe implicit Q theoremJacobi's eigenvalue methodNon-normalityOrthogonalityPerturbation theoryPseudospectrum