Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Line relaxation — where it appears

Solving simultaneously for a whole line of unknowns rather than one at a time. It is what repairs a point smoother on an anisotropic operator, at the cost of a tridiagonal solve per line and of knowing which direction is strong.

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

10⁻⁴10⁻³10⁻²10⁻¹100.250.50.751anisotropy εsmoothing factor μpoint + fully-line + fullpoint + semi-y⅓, the one-dimensional answertwo routes, three curvesgap between the repairs2.2·10⁻¹⁶scan against closed form2.2·10⁻¹⁶the dashed curve lies on the solid one beneath itone repair, written two ways

Smoothing a whole line at once

Solve every grid line in the strong direction exactly rather than sweeping over it, and the smoothing factor goes from 0.9993 back to 0.3340 — which is the one-dimensional answer, on a problem that is not one-dimensional. The repair replaces one ε in the closed form by a one.

iterative · Anisotropy 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

Named alongside it

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

AnisotropySmoothing factorBlock jacobiDiagonal dominanceFourier modesM-matrixProlongationRestrictionRotated anisotropySemi-coarseningTridiagonal solveThe V-cycle

All concepts