Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Short recurrence — where it appears

A recurrence whose next term needs only the last two, so the storage does not grow with the step number. It exists for a symmetric matrix and not for a general one, which is the single largest difference in cost between the symmetric and non-symmetric Krylov families.

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

10²10⁴10⁶10⁸10¹⁰0285684112140168196224condition numberstepsnormal equationsbidiagonalisationone sequence, two costssteps at κ = 10², both16at κ = 10⁶, ratio1.1at κ = 10¹⁰, ratio1.9the same iterates in the algebraand twice the work at κ = 10¹⁰

One sequence and two recurrences

CGLS and LSQR compute the same iterates — the minimiser over a space is unique, so there is nothing to choose between them in the algebra. At κ = 10⁶ they cost 42 steps and 47. At κ = 10¹⁰ they cost 110 and 209, across four seeds, and the quantity that separates them is the orthogonality of a basis neither of them keeps.

iterative · Krylov 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.

Krylov subspaceArnoldiBiorthogonalityCglsCounterexampleFinite terminationForward errorGolub kahan bidiagonalisationLanczosLook-aheadLsqrNormal equations

All concepts