Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Hermite interpolation — where it appears

Interpolation that matches a function's value and its derivative at each point. A two-sided projection of a linear model achieves it at every interpolation point, so r points buy 2r conditions rather than r.

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

10⁻¹110¹10²10³10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵s, on the real axis|H − Hᵣ| ÷ |H|exact where askedpoints4conditions bought8worst at a point5.3·10⁻¹⁶worst away from one8.4·10⁻⁴4 points, 8 conditionsand no bound in between

Exact at the points that were named

Balanced truncation asks for nothing and bounds everything, at a cost no large model can pay. The other kind of reduction asks for r numbers, costs r solves, is exact at every one of them — and bounds nothing anywhere else. That trade is the whole of large-scale model reduction.

reduction · moment matching 01234567891011121310⁻¹110¹10²10³iterationinterpolation point σa fixed point that is a conditionorder4iterations13σ against −λ(Aᵣ)2.8·10⁻¹²H₂ error4.3·10⁻⁵it stopped movingand the condition holds there

Interpolating at the model’s own poles

One choice of interpolation points is not arbitrary — the mirrored poles of the model about to be built. It is a fixed point rather than a guess, and when it is reached it beats a method costing O(n³) — by 0.4 per cent, which is the honest size of the whole contest.

reduction · moment matching

Named alongside it

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

Balanced truncationFlop countMoment matchingRational krylovTransfer functionA-priori boundCondition numberFixed point iterationH2 normKrylov subspaceLyapunov equationOptimality condition

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