Direct vs. iterative is certainly one of the key questions in linear solvers for sparse matrices as you have already observed. There are many misconceptions and lots of misinformation on this topic. So I encourage you to approach the issue from this point of view rather than looking for "5 methods."

Here is a nice introduction from one of the major contributors to direct methods:

[Matrix methods, Duff I.S.][1]

You refer to Professor Saad's classic book on iterative methods for sparse systems.
In the course he teaches at U. of Minnesota, he spends about equal time on direct and
iterative methods. His notes are a gold mine of information.

[Sparse Matrix Computations][2]


  [1]: http://www.numerical.rl.ac.uk/reports/duffRAL98076.pdf
  [2]: http://www-users.cselabs.umn.edu/classes/Spring-2014/csci8314/