A reliable modification of the cross rule for rational Hermite interpolation

Claessens' cross rule [8] enables simple computation of the values of the rational interpolation table if the table is normal, i.e. if the denominators in the cross rule are non-zero. In the exceptional case of a vanishing denominator a singular block is detected having certain structural properties so that some values are known without further computations. Nevertheless there remain entries which cannot be determined using only the cross rule.In this note we introduce a simple recursive algorithm for computation of the values of neighbours of the singular block. This allows to compute entries in the rational interpolation table along antidiagonals even in the presence of singular blocks. Moreover, in the case of non-square singular blocks, we discuss a facility to monitor the stability.