  Preconditioners are generally essential for fast convergence in the iterative
solution of linear systems of equations. However, the computation of a good
preconditioner can be expensive. So, while solving a sequence of many linear
systems, it is advantageous to recycle preconditioners, that is, update a
previous preconditioner and reuse the updated version. In this paper, we
introduce a simple and effective method for doing this. Although our approach
can be used for matrices changing slowly in any way, we focus on the important
case of sequences of the type $(s_k\textbf{E}(\textbf{p}) +
\textbf{A}(\textbf{p}))\textbf{x}_k = \textbf{b}_k$, where the right hand side
may or may not change. More general changes in matrices will be discussed in a
future paper. We update preconditioners by defining a map from a new matrix to
a previous matrix, for example the first matrix in the sequence, and combine
the preconditioner for this previous matrix with the map to define the new
preconditioner. This approach has several advantages. The update is entirely
independent from the original preconditioner, so it can be applied to any
preconditioner. The possibly high cost of an initial preconditioner can be
amortized over many linear solves. The cost of updating the preconditioner is
more or less constant and independent of the original preconditioner. There is
flexibility in balancing the quality of the map with the computational cost. In
the numerical experiments section we demonstrate good results for several
applications, in particular when using an algebraic multigrid preconditioner.
