  We present a new method for solving parametrized linear systems. Under
certain assumptions on the parametrization, solutions to the linear systems for
all parameters can be accurately approximated by linear combinations of
solutions to linear systems for a small set of fixed parameters. Combined with
either direct solvers or preconditioned iterative solvers for each linear
system with a fixed parameter, the method is particularly suitable for
situations when solutions to a large number of distinct parameters or a large
number of right hand sides are required. The method is also simple to
parallelize. We demonstrate the applicability of the method to the calculation
of the response functions in electronic structure theory. We demonstrate the
numerical performance of the method using a benzene molecule and a DNA
molecule.
