In minisymposium: Polynomial and Rational Eigenvalue ProblemsWed 12:00–12:30, Room AV 91.12
We propose a new set of basis polynomials for representing the Birkhoff interpolation polynomial, which is an extension of the well-known Lagrange and Hermite interpolation problems. The proposed basis extends the definition of the Newton basis for non-distinct interpolation nodes. This approach allows to determine the Birkhoff interpolation polynomial via a special linear system of equations. When applied to the special cases of Taylor, Lagrange and Hermite interpolations, this approach reduces to the well-known solutions of these problems expressed in the Newton basis. Therefore, the characteristics of the Birkhoff basis can be exploited from Newton basis. In particular, the structure of the polynomial as well as companion matrices will be as straightforward as for Newton basis.