Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Kronecker sum — where it appears

The sum of Kronecker products in which each term applies one factor and the identity in every other position. It is what a separable operator on a product domain is, its eigenvalues are all the sums of its factors' eigenvalues, and it stores d n-squared numbers rather than n to the 2d.

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

03672108144180216024681012eigenvalues in orderλthe closed formmarks: the assembled matrix, decomposeda spectrum nobody computedrows of the matrix216numbers that describe it108λ smallest0.59λ largest11worst |computed − exact|7.1·10⁻¹³the matrix is never neededand neither is its decomposition

An index that is a pair

A discretisation on a two-dimensional grid of n points a side has n² unknowns and a matrix with n⁴ entries — 10⁸ at n = 100. What that matrix is instead is two Kronecker products of an n × n matrix, which is 2n² numbers, and nothing has been approximated: assembling it was the mistake.

tensor · kronecker 024681012141610⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹index of the singular value at the cutσ ⁄ σ₁eight digits10^-4: 5 Kronecker terms10^-8: 7 Kronecker terms10^-12: 8 Kronecker termsnot closed, and nearly closedrank at the cut8a Kronecker product's1terms at 10⁻⁴5terms at 10⁻⁸7terms a decade0.5the inverse leaves the formatby half a term a decade

A solve that is d decompositions

A Kronecker sum is closed under nothing useful — its inverse is not a Kronecker sum and no factorisation of it is one. What it has instead is eigenvectors that are Kronecker products, so a solve with 1,728 unknowns takes one decomposition of a 12 × 12 matrix and nothing else.

tensor · kronecker 081624324010⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²conjugate gradient steprelative residualabove, on their own scale: rank asked 5, rank kept 3solid: the budgeted residual · light: no budgetbudget 3rank asked for5rank kept3residual, budgeted3.8·10⁻⁴residual, unbudgeted1.6·10⁻¹⁴numbers stored150the step asks for moreat every step

An iterate that must be made smaller

Applying a Kronecker-sum operator to a low-rank iterate multiplies its ranks by d and adding two of them adds their ranks, so a solver in a compressed format cannot keep what it produces. Every step is followed by a truncation — and whether that truncation is a floor on the residual depends on the right-hand side rather than on the truncation.

iterative · low rank iteration

Named alongside it

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

Model problemCondition numberCurse of dimensionalityDiscrete laplacianExact ground truthKronecker productSeparabilityConjugate gradientsFlop countKrylov subspaceLow-rank approximationMatrix-free

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