  This paper presents a parallel algorithm for finding the smallest eigenvalue
of a particular form of ill-conditioned Hankel matrix, which requires the use
of extremely high precision arithmetic. Surprisingly, we find that
commonly-used approaches that are designed for high efficiency are actually
less efficient than a direct approach under these conditions. We then develop a
parallel implementation of the algorithm that takes into account the unusually
high cost of individual arithmetic operations. Our approach combines message
passing and shared memory, achieving near-perfect scalability and high
tolerance for network latency. We are thus able to find solutions for much
larger matrices than has been previously possible, with the potential for
extending this work to systems with greater levels of parallelism.
