The following basic algorithms for sparse eigenproblems will be included.
This list is not exhaustive, and we are actively looking for other algorithms. Also, some common methods may classified in several ways. For example, simultaneous iteration and block Arnoldi with an immediate restart are identical. These categories are not meant to be mutually exclusive, but to be helpful to the user. We will include some older but commonly used methods, just to be able to advise the user to use more powerful alternatives, and for experimental purposes.
Arnoldi and Lanczos methods are Krylov subspaces based techniques.
These methods may converge very fast in combination with
shifted-and-inverted operators, which means that 
has
to be used in matrix vector products in each iteration step. If only
approximations for are available then Davidson's
method can be used as an acceleration technique for the inexact
shift-and-invert operations. Approximations for
can be computed from a preconditioner for 
or by a few steps
of a (preconditioned) iterative method [35].
Trace minimization is suitable for self-adjoint
problems, and uses optimization techniques like conjugate
gradients to find the 
smallest eigenvalues.
There is unfortunately no simple way to identify the best algorithm and choice of options to the user. The more the user discovers about the problem (such as approximate eigenvalues), the better a choice can be made. In the common situation where the user is solving a sequence of similar problems, this is quite important.
There is also unfortunately no inexpensive way to provide a
guarantee that all eigenvalues in a region have been found,
when the problem is not self-adjoint.
For Hermitian eigenvalue problems by factoring certain translations of 
by the identity, it is possible to guarantee that all eigenvalues in a
region have been found. In the non-Hermitian case, this same task is
accomplished by a vastly more expensive technique called a Nyquist plot
(i.e. compute the winding number).
However, depending on the
problem, there are methods to help increase one's confidence
that all eigenvalues have been found.