Multivariate Rational Approximation Using a Stabilized Sanathanan-Koerner Iteration

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.