Circulant preconditioner — where it appears
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
Also named here as clustered spectrum — the same set of essays touches all of them, so they are one junction rather than several.
A preconditioner that changes sign
Strang's circulant preconditioner takes Toeplitz conjugate gradients from 179 steps to 10 at n = 256. At n = 64 on the same family it takes 66 steps to 109 — worse than doing nothing. Between those rows the preconditioner's smallest eigenvalue crosses zero, and nothing in the published account of the method mentions that it can be negative.
The circulant that cannot be indefinite
The previous essay found a preconditioner taking 117 steps against an unpreconditioned 59, because its smallest eigenvalue was −0.173. Average the two diagonals instead of choosing between them and the count is 7, 8, 9, 10, 10 across a factor of sixteen in size.
Two dimensions, and the cluster that thins
The same kernel, the same averaging, the same transform — applied along two axes instead of one. In one dimension the preconditioned step count is 7, 10, 10, 10; on square grids with the same unknown counts it is 10, 18, 20, 21, and the share of the spectrum near one falls from 56% to 17%.
Named alongside it
The objects these essays reach for when they reach for this one.
Clustered spectrumConjugate gradientsPreconditioningToeplitz matrixCondition numberDiscrete fourier transformPositive definiteAsymptotic analysisCirculant matrixConvergence rateFrobenius normSymbol