**``Dir''**- is direct application, where we perform a matrix multiply on a vector in each step. It is the simplest variant to apply, since the matrix can be stored in any compact way. On the other hand, most algorithms need many matrix-vector multiplications to converge and are restricted to seeking eigenvalues at the ends of the spectrum.
**``SI''**- is the shift-and-invert which needs a factorization routine to enable solutions of systems for , but gives the ability to compute a wider choice of eigenvalues in fewer iterations.
**``Prec''**- means application with a preconditioner, for instance, a sparse approximate factorization. This requires less space than the shift-and-invert, but most often it also needs a larger number of matrix-vector multiplies.

Susan Blackford 2000-11-20