Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Multiplicity — where it appears

How many times an eigenvalue is repeated, and the dimension of the eigenspace a method must return that many vectors from.

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

λ = 105 timesλ = 9.55 timesλ = 95 timesλ = 8.55 timesλ = 2.952 timesλ = 2.92 timeseigenvalues that arrived more than once — the matrix has 40 distinct onesa spectrum with the wrong multiplicitiesextra copies, no reorthogonalisation25extra copies, full reorthogonalisation0worst relative error among the copies1.9·10⁻⁸steps taken of 80 asked for, full40no arithmetic error was madeevery one of these is right to eight digits

An eigenvalue that arrives twice

A matrix with forty distinct eigenvalues, handed to Lanczos for eighty steps, returns twenty-five extra copies of thirteen of them — the largest arriving five times. Every copy is accurate to 1.9·10⁻⁸ relative. No arithmetic error was made, nothing overflowed, and a caller counting eigenvalues gets the wrong multiplicity from a computation in which no individual number is wrong.

spectra · lanczos 12345610⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹Ritz value, largest firstdistance from 10one vectora block of twoa space, not a ratecopies found, one vector1copies found, block of two2Krylov dimension, one vector16the second copy is not in the spaceat any number of steps

An eigenvalue one vector cannot see

A matrix with an exactly doubled eigenvalue at 10. Twelve Lanczos steps find it once; twenty-four find it once, on a Krylov space of dimension 23 in a 24-dimensional problem. A block of two vectors finds it twice. This is not slow convergence — the second copy is not in the space.

spectra · invariant subspace

Named alongside it

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

Krylov subspaceLanczos algorithmRitz valuesBlock methodsEigenvalue gapEigenvaluesEigenvector conditioningInvariant subspaceOrthogonalityReorthogonalisationThree term recurrence

All concepts