Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Spurious spectrum — where it appears

The eigenvalues a linearised approximation returns that are not eigenvalues of the problem it approximates. They are exactly right about the approximant, so nothing in the arithmetic marks them, and telling them apart needs the original function.

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

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 -10123456789-101λbranch point at −0.4the closed formwhich of these is an answereigenvalues returned36wanted6past the branch point6complex24worst against the closed form7.7·10⁻⁴all of them exactfor a problem nobody asked

The eigenvalues that are answers to nothing

A rational approximant of degree five turns a six-by-six problem into a thirty-six-by-thirty-six one, and thirty-six numbers come back. Six are the answer. The rest are exact eigenvalues of the approximant, lying where the function it approximates is not a real number at all.

polynomial · spurious spectrum

Named alongside it

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

Approximation before linearisationBranch pointLinearisationRational approximationChebyshev basisCompanion formComrade matrixCondition numberEigenvalueExact ground truthFlop countNonlinear eigenvalue problem

All concepts