Numerical Linear Algebra
where the algebra stops being the arithmetic
Series

Hyperbolic quadratic — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. -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.

    part 1 · polynomial
  2. 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.

    part 2 · polynomial
  3. 8 massescompanion, ε = 10⁻¹²2.7·10⁻⁹definite, ε = 10⁻¹²0.00210⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²ε, the margin above critical dampingworst relative error of the spectrumcompaniondefinite pencilthe margin shrinks to the leftthe guarantee costs the digits

    A guarantee paid for in digits

    A hyperbolic quadratic has a real spectrum, but the route that computed it so far, a companion linearisation and a nonsymmetric Schur form, only happens to return one. The theory offers a route on which it cannot do otherwise: a symmetric linearisation that is a definite pencil, reduced by a Cholesky to a symmetric eigenproblem. Near the class's boundary that guarantee is the expensive route. On a damped chain at damping β*(1 + ε) the definite pencil's Crawford number — what its Cholesky divides by — vanishes like 0.228ε, every combination of the pencil shrinking with it, while the colliding pair the companion route has to resolve separates like 0.98√ε. At ε = 10⁻¹² the definite route keeps two or three digits and the companion route eight or nine. And on twenty rotated chains the companion route returns no complex pair at any margin from 10⁻⁴ to 10⁻¹², only on the boundary itself.

    part 3 · polynomial

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