The convergence behavior depends to a great deal on the distribution of the spectrum of . Two matrices with precisely the same sparsity pattern could exhibit completely different convergence characteristics for a given choice of and . Therefore, it would be misleading to draw any conclusions based solely on operation counts per iteration. Nevertheless, it can be useful to have a notion of the major costs of an iteration to guide initial choices.
In the following, we assume that the cost of a matrix-vector product is . For a sparse matrix, one could think of as twice the average number of nonzero entries per row of . For a dense matrix , and for a fast Fourier transform (FFT) matrix . See §10.2. For a fixed value of and , the cost (in floating point operations) for one iteration of IRLM is
If the matrix-vector product is cheap (i.e., is quite small), the cost of implicit restart is significant and alternatives should be considered. One of these might be to use a polynomial spectral transformation. This would involve operating with a suitably constricted polynomial function of in place of . An example would be a Chebyshev polynomial designed to damp the unwanted portion of the spectrum. The action would, of course, be applied as a succession of matrix-vector products involving .