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.

Named by 2 essays across 2 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 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 iterationCglsEigenvectorsFilter factorsForward errorGMRESGolub kahan bidiagonalisationIll posed problemNon normal matrixNormal equationsOrthogonality loss

All concepts