Concept

Double root — where it appears

An eigenvalue of multiplicity two, where two roots of the same polynomial coincide. The map from a coefficient to such a root has an infinite derivative, so a perturbation of size ε moves the root by the square root of ε and a computation in binary64 returns about eight digits rather than sixteen.

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

Also named here as hyperbolic quadratic, overdamping — the same set of essays touches all of them, so they are one junction rather than several.

Every eigenvalue real, and a test that says so

A quadratic eigenvalue problem has no reason to have real eigenvalues. One class does, as a property rather than an outcome, and the proof is a Cholesky that completes. The boundary of the class has a closed form, and at the boundary the arithmetic loses half its digits with nothing ill conditioned anywhere.

polynomial · Hyperbolic quadratic

A class a longer chain takes away

Symmetry survives a bigger problem. Hyperbolicity does not. The damping that certifies a chain of seven masses is refused by a chain of eight, the damping the class demands grows like the length without bound, and the certificate has to be earned again at every size — which costs one Cholesky, and the alternative is a proof quietly inherited from a smaller problem.

polynomial · Hyperbolic quadratic

Named alongside it

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

Cholesky Condition number Exact ground truth Hyperbolic quadratic Inertia Overdamping Quadratic eigenvalue problem Defective matrix

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