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In addition to the three kernels above, the iterative methods using shift-and-invert often require the direct solutions of the linear systems. Both matrix factorization and triangular solve involve much more complicated parallel algorithms, especially on massively parallel machines. There has been a large amount of research activity in this area. Many state-of-the-art parallel algorithms for dense and band matrices are implemented in ScaLAPACK (see § 10.3), and those for sparse matrices are implemented in the software packages surveyed in Table 10.1.

Susan Blackford 2000-11-20