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{\large Paul G. Constantine \\
{\bf A Stieltjes-Lanczos Method for Parameterized Matrix Equations}}

PO BOX 5800 \\ MS 1318 \\ Albuquerque \\ NM 87185
\\
{\tt pconsta@sandia.gov}\\
David F. Gleich\end{center}

We develop a Stieltjes-Lanczos procedure for a symmetric, positive
definite parameterized matrix $A(s)$ and a given
parameterized vector $b(s)$. Each element in $A(s)$ and $b(s)$ is assumed
to be a bounded and continuous
function of a set of parameters $s\in\mathcal{D}\subset\mathbb{R}^d$.
The method computes a \emph{constant} tridiagonal matrix whose
eigenvalues approximate the parameterized spectrum of
$A(s)$. We show how this can be interpreted as constructing a Gaussian
quadrature formula for a Riemann integral of the
form $\langle b^Tf(A)b \rangle$, where $f(\cdot)$ is an analytic function
and $\langle\cdot\rangle$ denotes
integration over the parameter space.
We also apply this procedure to iteratively approximate the vector-valued
function $x(s)$ that solves
$A(s)x(s)=b(s)$; 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.


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