A general recurrence interpolation formula and its applications to multivariate interpolation

Elimination techniques: from extrapolation to totally positive matrices and CAGD

“…From the variation diminution produced by the totally positive matrices of the process, it can be deduced that the curve imitates better the form of the control polygon B 0 • • • B n than that of the control polygon P 0 • • • P n . Therefore, we see again that an NTP basis (b 0 ; : : : ; b n ) of a space U has optimal shape-preserving properties if for any other NTP basis (p 0 ; : : : ; p n ) of U there exists a (stochastic) TP matrix K such that (p 0 ; : : : ; p n ) = (b 0 ; : : : ; b n )K: (19) Hence, a basis has optimal shape preserving properties if and only if it is a normalized B-basis. Neville elimination has also inspired the construction of B-bases in [11,12].…”

“…Some papers, [38,46] among others, extended both approaches at the beginning of the 1970s, to the more general setting of Chebyshev systems. Almost simultaneously, extrapolation methods were being studied and extended by several authors, as Schneider [54], Brezinski [4,5,7], H avie [31][32][33], M uhlbach [39 -42,48] and Gasca and LÃ opez-Carmona [19]. For a historical overview of extrapolation methods confer Brezinski's contribution [6] to this volume and the book [8].…”

Two composition methods for solving certain systems of linear equations

“…The general case seems to constitute a new method for solving systems of linear equations based on the solutions of certain of its subsystems 9 where different proofs can be found. Theorem 2 with m = 1, k = n-1 can also be derived from the General Newton algorithm [14,6]. Theorem 2 as given here combines and extends these algorithms giving also necessary and sufficient conditions for the regularity of A.…”

A unified approach to evaluation algorithms for multivariate polynomials

“…To these points, we shall add three more points: 21 . These are precisely the n+2 2 points that give rise to Lagrange interpolation conditions.…”

“…Both the divided and forward difference algorithms are discussed in standard numerical analysis textbooks [10]. We refer the readers to [32,21] for a generalization of divided differences to the multivariate setting. We now describe how a bivariate version of the divided difference and forward difference algorithms can be obtained to evaluate a bivariate polynomial along a grid {(a j , b k )}.…”