We develop a Stieltjes-Lanczos procedure for a symmetric, positive
definite parameterized matrix and a given
parameterized vector
. Each element in
and
is assumed
to be a bounded and continuous
function of a set of parameters
.
The method computes a constant tridiagonal matrix whose
eigenvalues approximate the parameterized spectrum of
. We show how this can be interpreted as constructing a Gaussian
quadrature formula for a Riemann integral of the
form
, where
is an analytic function
and
denotes
integration over the parameter space.
We also apply this procedure to iteratively approximate the vector-valued
function
that solves
; such problems commonly arise within discretizations of
partial differential equations with
stochastic inputs. Preliminary numerical experiments are provided which
validate the theory and suggest future
directions for research.