Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Palindromic quadratic — where it appears

A quadratic matrix polynomial that reads the same backwards, with the same coefficient at both ends and a symmetric one between. Its eigenvalues come in pairs whose products are one, which says that a wave in a periodic structure travels equally well both ways.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Also named here as reciprocal pair, structure preserving — the same set of essays touches all of them, so they are one junction rather than several.

-5-3.20611-1.412210.3816782.175573.969460log₁₀ |λ||λ| = 1λλ′ = 1, in the algebrapairs6decades of spectrum9.6pairing error, general3.3·10⁻⁹pairing error, structured2.2·10⁻¹⁶a symmetry the solver never knew aboutand the half of the answer it decides

A spectrum that comes in reciprocal pairs

A palindromic quadratic reads the same backwards, so λ is an eigenvalue exactly when 1/λ is. A general solver discards that, computes the large half of the spectrum perfectly and the small half to seven digits — and the small half is a division away from being perfect too.

polynomial · structured spectrum 1357910⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴log₁₀ b, the couplingrelative errorsmall half, directstructure-preservingsmall half, by 1/λone divisionlarge half, direct5.5·10⁻¹⁵small half, direct10⁻⁷small half, by 1/λ4.9·10⁻¹⁵structure-preserving6.5·10⁻⁶the structure is not decorationit is where half the accuracy is

A perturbation that keeps the symmetry

The smallest perturbation that makes a computed answer exact is the backward error. Ask for the smallest one that also keeps the problem's structure and the number can only go up — and measured on a palindromic quadratic it goes up by 1.17, while the structure the computed spectrum has lost is not in either number.

structure · structured backward error

Named alongside it

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

Quadratic eigenvalue problemReciprocal pairStructure preservingStructured backward errorBackward errorCayley transformGeneralised eigenvalue problemPerturbationRelative accuracy

All concepts