Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Target set — where it appears

The region a caller states an approximation must be good on, chosen before anything is computed. How close it comes to a singularity sets the rate at which every approximant converges, and it cannot be revised once the approximation is built.

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

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 263850627410⁻⁵10⁻³10⁻¹rows in the linearisation‖g − r‖ on the target setpolynomialrationalequal cost, two basesreach0.06largest size drawn78rational there2·10⁻⁵polynomial there0.0015the ratio73no winner on an easy targetand two orders on a hard one

Two approximants and one matrix size

A polynomial approximant linearises to nd rows and a rational one to n(m+1), so the fair contest fixes the matrix and varies the basis. On an easy target set the two are indistinguishable and the ordering flips with the noise; on one that reaches a branch point the rational pulls away by two orders.

polynomial · approximant choice

Named alongside it

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

Approximation before linearisationBranch pointCondition numberRational approximationChebyshev basisCompanion formComrade matrixFlop countLeast squaresLinearisationNonlinear eigenvalue problemShift selection

All concepts