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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