Previous: Summary of Choices.
For each algorithm we have
distinguished between different ways of preparing the matrix
prior to running the algorithm.
- 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.
- 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.
- 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.