  This paper introduces and explores the class of expansion preconditioners
EX(m) that forms a direct generalization to the classic complex shifted Laplace
(CSL) preconditioner for Helmholtz problems. The construction of the EX(m)
preconditioner is based upon a truncated Taylor series expansion of the
original Helmholtz operator inverse. The expansion preconditioner is shown to
significantly improve Krylov solver convergence rates for the Helmholtz problem
for growing values of the number of series terms m. However, the addition of
multiple terms in the expansion also increases the computational cost of
applying the preconditioner. A thorough cost-benefit analysis of the addition
of extra terms in the EX(m) preconditioner proves that the CSL or EX(1)
preconditioner is the practically most efficient member of the expansion
preconditioner class. Additionally, possible extensions to the expansion
preconditioner class that further increase preconditioner efficiency are
suggested.
