  The Sanathanan-Koerner iteration developed in 1963 is classical approach for
rational approximation. This approach multiplies both sides of the
approximation by the denominator polynomial yielding a linear problem and then
introduces a weight at each iteration to correct for this linearization.
Unfortunately this weight introduces a numerical instability. We correct this
instability by constructing Vandermonde matrices for both the numerator and
denominator polynomials using the Arnoldi iteration with an initial vector that
enforces this weighting. This Stabilized Sanathanan-Koerner iteration corrects
the instability and yields accurate rational approximations of arbitrary
degree. Using a multivariate extension of Vandermonde with Arnoldi, we can
apply the Stabilized Sanathanan-Koerner iteration to multivariate rational
approximation problems. The resulting multivariate approximations are often
significantly better than existing techniques and display a more uniform
accuracy throughout the domain.
