We propose an algorithm called BlockCGSI to compute some partial spectral information related to the ill-conditioned part of a given coefficient matrix. This information can then be used to improve the solution of consecutive linear systems with the same coefficient matrix and changing right-hand sides.
The BlockCGSI algorithm combines the block Conjugate Gradient with the inverse Subspace Iteration. We indicate how to reduce the total amount of computational work by controlling appropriately the accuracy when solving the linear systems at each inverse iteration. We also improve the global convergence of the algorithm by means of polynomial filters.