  We provide a comparative study of the Subspace Projected Approximate Matrix
method, abbreviated SPAM, which is a fairly recent iterative method to compute
a few eigenvalues of a Hermitian matrix $A$. It falls in the category of
inner-outer iteration methods and aims to save on the costs of matrix-vector
products with $A$ within its inner iteration. This is done by choosing an
approximation $A_0$ of $A$, and then, based on both $A$ and $A_0$, to define a
sequence $(A_k)_{k=0}^n$ of matrices that increasingly better approximate $A$
as the process progresses. Then the matrix $A_k$ is used in the $k$th inner
iteration instead of $A$.
  In spite of its main idea being refreshingly new and interesting, SPAM has
not yet been studied in detail by the numerical linear algebra community. We
would like to change this by explaining the method, and to show that for
certain special choices for $A_0$, SPAM turns out to be mathematically
equivalent to known eigenvalue methods. More sophisticated approximations $A_0$
turn SPAM into a boosted version of Lanczos, whereas it can also be interpreted
as an attempt to enhance a certain instance of the preconditioned
Jacobi-Davidson method.
  Numerical experiments are performed that are specifically tailored to
illustrate certain aspects of SPAM and its variations. For experiments that
test the practical performance of SPAM in comparison with other methods, we
refer to other sources. The main conclusion is that SPAM provides a natural
transition between the Lanczos method and one-step preconditioned
Jacobi-Davidson.
