An algorithm for two-variable rational interpolation is developed. The
algorithm is suitable for interpolation cases where neither the number
of interpolation points to be used nor the final degrees of the rational
interpolant are known a priori. Instead, a maximum degree for the
interpolant's numerator and denominator is assumed, and, by testing the
condition number of the interpolation system's matrix at each step, the
necessary reductions are made so as to cope with non-normality and
unattainability occasions. The algorithm can be used for applications of
the Evaluation-Interpolation technique in matrix manipulations, such as
finding the inverse of a matrix with elements rational functions in two
variables. The algorithm avoids completely symbolic calculations, thus
keeping the execution time very low even if the system size is large,
and achieves accurate function recoveries for greater polynomial degrees
than other bivariate rational interpolation methods.