  A widely used approach to compute the action $f(A)v$ of a matrix function
$f(A)$ on a vector $v$ is to use a rational approximation $r$ for $f$ and
compute $r(A)v$ instead. If $r$ is not computed adaptively as in rational
Krylov methods, this is usually done using the partial fraction expansion of
$r$ and solving linear systems with matrices $A- \tau I$ for the various poles
$\tau$ of $r$. Here we investigate an alternative approach for the case that a
continued fraction representation for the rational function is known rather
than a partial fraction expansion. This is typically the case, for example, for
Pad\'e approximations. From the continued fraction, we first construct a matrix
pencil from which we then obtain what we call the CF-matrix (continued fraction
matrix), a block tridiagonal matrix whose blocks consist of polynomials of $A$
with degree bounded by 1 for many continued fractions. We show that one can
evaluate $r(A)v$ by solving a single linear system with the CF-matrix and
present a number of first theoretical results as a basis for an analysis of
future, specific solution methods for the large linear system. While the
CF-matrix approach is of principal interest on its own as a new way to compute
$f(A)v$, it can in particular be beneficial when a partial fraction expansion
is not known beforehand and computing its parameters is ill-conditioned. We
report some numerical experiments which show that with standard preconditioners
we can achieve fast convergence in the iterative solution of the large linear
system.
