Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Tropical roots — where it appears

The corners of the max-plus polynomial built from the norms of a matrix polynomial's coefficients, offered as an estimate of where its eigenvalues lie. For a quadratic they are ‖K‖/‖C‖ and ‖C‖/‖M‖ when ‖C‖² exceeds ‖M‖‖K‖, and √(‖K‖/‖M‖) twice when it does not. Three norms and a comparison compute them, and the standard scaling of a matrix polynomial is built from the same three numbers.

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

0246810⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹log₁₀ γ, the change of unitsforward erroras writtenafter scalingone change of variableunscaled, worst0.0013scaled, worst1.7·10⁻¹³orders recovered10scaled coefficient spread4.5the answer was never the problemthe units were

The scaling that buys ten orders

Two lines computed from three norms, a change of variable that is exact in both directions, and the whole of the loss the previous essay measured comes back — flat, at every stop, because after scaling every stop is the same problem.

polynomial · Polynomial scaling 11.31.610⁻¹110¹10²10³log₁₀ ntropical root ÷ actual modulusexactsmallest moduluslargest modulusthree norms, two endslarge root, n = 40.96large root, n = 320.94small root, n = 45.4small root, n = 32230a maximum predicts a maximumand says nothing about a minimum

An estimate that does not move

The tropical roots are said to miss the bottom of a spectrum by a factor growing like n². The estimate does not get worse. It changes by four per cent between four masses and sixty-four while the modulus it names falls by a factor of a hundred and sixty, and an estimate that is constant in the variable the answer depends on is a different defect from an inaccurate one.

polynomial · Polynomial scaling

Named alongside it

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

Exact ground truthMatrix polynomialQuadratic eigenvalue problemScalingA-priori boundAsymptotic analysisBackward errorCondition numberFrobenius normLinearisation

All concepts