Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Lsqr — where it appears

Least squares through a Golub–Kahan bidiagonalisation and a plane rotation, computing the same iterates as CGLS in exact arithmetic and not in floating point. It is the method to use on a least-squares problem too large to factorise, and its equivalence to CGLS holds in exact arithmetic and visibly fails in floating point.

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

036912151821242700.250.50.7511.25index kweight appliedonebidiagonalArnoldiis it a function of σmisfit, even fit1.6·10⁻¹²misfit, general fit0.063‖A − Aᵀ‖/‖A‖0.086one method's weights do not notice the operatorand the other's stop being a function of σ

The basis decides what a filter is

The vocabulary of regularisation is spectral — a method keeps a component or discards it, and the weights are a function of the singular value. Row-normalising a symmetric blur so that it preserves a constant makes it 8.6% asymmetric, and that is enough to move GMRES's weights from 7·10⁻¹⁴ off a function of σ to 4.4·10⁻².

regularisation · GMRES 10²10⁴10⁶10⁸10¹⁰110²10⁴10⁶10⁸10¹⁰condition number of the matrixcondition number seen by the iterationκ(A), unchangedκ(AR⁻¹)a bound with no κ(A) in itκ(AR⁻¹), every κ(A)2.2κ(SU), the other route2.2κ(A), across the sweep10⁸κ(AR⁻¹), across the sweep1the sketch never sees the spectrumand the spectrum cancels out of the answer

The sketch that is not the answer

Sketch-and-solve throws away the original problem and keeps the small one's answer, which is why its answer moves with the seed. Use the same sketch as a preconditioner instead and the condition number the iteration sees is the same number at every κ from a hundred to ten billion — identically the same, to nine digits, because the spectrum cancels out of it.

randomised · Sketching 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

Named alongside it

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

Krylov subspaceArnoldi iterationCglsCondition numberEigenvectorsFilter factorsForward errorGMRESGolub kahan bidiagonalisationIll-posed problemMatrix-freeNon-normal matrices

All concepts