Bivariate Thiele-type matrix-valued rational interpolants

A new method for the construction of bivariate matrix-valued rational interpolants on a rectangular grid is introduced in this paper. The rational interpolants are of the continued fraction form, with scalar denominator. In this respect the approach is essentially different from that of Bose and Basu (1980) where a rational matrix-valued approximant with matrix-valued numerator and denominator is used for the approximation of a bivariate matrix power series. The matrix quotients are based on the generalized inverse for a matrix introduced by Gu Chuanqing and Chen Zhibing (1995) which is found to be effective in continued fraction interpolation. A sufficient condition of existence is obtained. Some important conclusions such as characterisation and uniqueness are proven respectfully. The inner connection between two type interpolating functions is investigated. Some examples are given so as to illustrate the results in the paper.