Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Lanczos — where it appears

The three-term recurrence generating an orthogonal basis of a Krylov space of a symmetric matrix, whose orthogonality is lost long before its accuracy is. It is what makes a symmetric eigensolve affordable at scale, and its loss of orthogonality is a collapse rather than a drift — it begins when the first eigenvalue converges.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

1234567891010⁻⁵⁵10⁻⁴⁹10⁻⁴³10⁻³⁷10⁻³¹10⁻²⁵10⁻¹⁹10⁻¹³10⁻⁷10⁻¹cyclesizethe residualrepairedthe reported boundwhat the stopping rule readsreported at the last cycle9.4·10⁻⁴¹the residual there5.7·10⁻⁵with the border recomputed3.8·10⁻¹⁴products, cheap and repaired9a bound with nothing under itand one product a cycle to fix it

Keeping the vectors, and losing the bound

Thick restarting keeps the Ritz vectors instead of filtering the starting vector — the same eigenvalues for a third of the products with A. Its residual bound reaches 9.4·10⁻⁴¹ while the residual it bounds sits at 5.7·10⁻⁵, and the eigenvalues are correct to 4.3·10⁻¹⁴ the whole time, so nothing reports it.

spectra · Lanczos 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 ran to the end3173lucky — a subspace closed295serious, cured by a block of two495serious, cured by a longer block29serious, incurable at any length8counted, not estimatedserious, as a fraction0.13of those, cured at two0.93incurable8matrices tried4000measure zero on the realsand an eighth of the integers

The same zero, and nothing was found

Change the recurrence by two lines and the divisor stops being a norm. It becomes an inner product of two vectors from two different sequences, and an inner product of two different vectors is zero on a whole hyperplane — with neither vector anywhere near zero, nothing invariant, and nothing converged. The arithmetic event is identical and the meaning is opposite.

iterative · Breakdown

Named alongside it

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

Invariant subspaceKrylov subspaceReorthogonalisationRitz valuesArnoldiArrowhead matrixBiorthogonalityBlock methodsCommunicationCounterexampleDeflationFinite termination

All concepts