Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Overdamping — where it appears

The regime of a damped system in which every mode returns to rest without oscillating, so no eigenvalue has an imaginary part. For a chain whose damping is proportional to its stiffness it begins at the critical value 1/sin(π/2(n+1)), which grows with the number of masses without bound.

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

-33-28.2505-23.5009-18.7514-14.0018-9.25229-4.502750eigenvalueQ(μ) ≺ 0one Cholesky, two answersabove the certificate8below it8the gap0.67critical β for this n5.8the spectrum is real by classnot by outcome

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 22.539343.078693.618034.157384.6967210⁻¹110¹10²stiffness damping βoverdamping marginβ* = 3.23607two routes to a boundaryclosed form β*3.2by certificate3.2difference4.7·10⁻¹³bisection steps44a factorisation that completesand a sine, agreeing to twelve digits

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.

CholeskyCondition numberDouble rootExact ground truthHyperbolic quadraticInertiaQuadratic eigenvalue problemDefective matrix

All concepts