Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

M matrix — where it appears

2 essays name this object, across one field. What follows is each of them, and the objects they name alongside it.

Also named here as semi coarsening — the same set of essays touches all of them, so they are one junction rather than several.

point · strong in y0.951point · strong in x0.956y-line · strong in y0.037y-line · strong in x0.967semi-y · strong in y0.107semi-y · strong in x0.9661.00 — no convergenceone problem, seen from two sidesy-line, aimed0.037y-line, turned sideways0.97the ratio2631×31 grid, twelve V-cyclesthe direction is in the code, not in the problem

Coarsening in one direction only

Leave the smoother alone and halve only the strong direction, and the smoothing factor is 0.3340 — identical to line relaxation's, at every anisotropy and every weight, to twelve digits. The convergence factors are then a factor of three apart, and at 45° both repairs fail outright.

iterative · anisotropy the grid the operator came froman edge is a coupling the matrix calls strongkeptinterpolatedwhat the entries decidedcoupling ratio, x against y0.001strong couplings across x0strong couplings along y110rows kept or dropped whole1111×11 grid, θ = 0.25the coarse grid, from the matrix alone

The coarse grid the matrix chooses

Given a tridiagonal matrix and no information about a grid, the coarsening keeps every other point and derives the weights ½, 1, ½ — the operators the geometric method was handed. Given the anisotropic operator, it discovers semi-coarsening, in the right direction, without a coordinate.

iterative · algebraic multigrid

Named alongside it

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

AnisotropyProlongationSemi coarseningAlgebraic multigridCoarse fine splittingGalerkin coarse operatorInterpolation weightsLine relaxationRestrictionRotated anisotropySmoothing factorStrength of connection

All concepts