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. A single starting vector cannot see more than one direction of a repeated eigenvalue, so detecting it is what a block method is for.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

-101-101(x − 1) ÷ 0.001p(x) ÷ 2.68·10⁻¹⁵a square, drawn negativepoints sampled401negative, unfused179negative, fused196signs disagree at98widest gap1.4·10⁻¹⁵unfusedfusedthe true valuethe true curve is the flat oneeverything else is rounding

A square that evaluates negative

(x − 1)⁶ evaluated near x = 1 comes out negative at 179 of 401 points on one build and 196 on another, and the two disagree about the sign at 98 of them. Neither is nearer the truth: both traces are made entirely of rounding.

machine · Fma contraction 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 an eigenvalue repeated 2 times, in a 200-dimensional problemblock of 1never returns them allblock of 232 products, 16 stepsblock of 351 products, 17 stepsblock of 460 products, 15 stepsblock of 570 products, 14 stepsblock of 684 products, 14 stepsthe narrowest that workscheapest width2its products with A32the widest drawn84narrower than the multiplicity is not slowit is absent

How wide the block should be

A block narrower than the multiplicity does not converge slowly — it never returns the missing copy at all. Above the multiplicity every extra column buys iterations at about ten products with A each. And the mechanism that is supposed to make the choice unimportant never fires from a random start.

spectra · Invariant subspace 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

Named alongside it

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

Krylov subspaceRitz valuesBlock methodsInvariant subspaceLanczos algorithmReorthogonalisationBisectionBitwise reproducibilityCatastrophic cancellationCommunicationDeflationEigenvalue gap

All concepts