Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Shift selection — where it appears

Choosing the points at which a method takes shifted solves, whether as ADI shifts, interpolation points or the poles of an approximant. It costs nothing to decide and is worth orders, and the answer is nearly always geometric clustering towards wherever the problem is difficult.

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

10¹10²10³10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹λ, over the spectrum of −A|r(λ)|geometric maxequally spacedgeometricone line of codeshifts6κ of the spectrum389geometric max0.12equally spaced max0.89the factor between7.5the same k solvesand one choice of where

Where to put the poles of a rational function

Three times in one field the same question arrives from different directions — ADI shifts, rational approximation of a square root, the decay of a Gramian — and it has one answer. Cluster them geometrically towards wherever the function is difficult, and the alternative that looks reasonable costs orders.

reduction · gramian decay 110¹10⁻⁷10⁻⁵10⁻³distance from the branch point, λ + c|g − r|the eigenvaluesdegree 98 poleschosen, not computedreach0.06left end, from the cut0.2rational, worst8.5·10⁻⁴polynomial, worst0.0068linearisation size, both54committed before the solveand invisible to it

An error committed before the arithmetic

Before a nonlinear eigenvalue problem is solved, somebody says where they think the eigenvalues are. That sentence sets the accuracy of everything that follows by five orders, costs nothing to say, and cannot be revised once the approximation built on it is in hand.

polynomial · approximation before linearisation

Named alongside it

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

Condition numberRational approximationZolotarev numberADI iterationApproximation before linearisationBranch pointEquioscillationGramianLeast squaresLow-rank approximationLyapunov equationNonlinear eigenvalue problem

All concepts