Generalized Schur-complements and a test for total positivity

New vector sequence transformations

“…Replacing a k by its expression in (24) (25) and (26) are similar to the rules of the E-algorithm [5]. Due to the non-commutativity of the matrix products, it does not seem that the preceding rules could be simplified.…”

“…19-23] or [42,Section 6.4] for basic definitions and results). Schur complements have many applications in extrapolation methods and related topics [7,8,13] and in other domains of mathematics, in particular in linear algebra [23,24,26,28,34].…”

Elimination techniques: from extrapolation to totally positive matrices and CAGD

“…This concept, introduced in [34,35] has found many applications in Linear Algebra and Statistics [13,53]. It may be generalized in di erent ways, see, for example, [21,22,44] where we used the concept of general elimination strategy which is explained in the next section.…”

“…However, it is clear that the basic role in all these papers was played by elimination techniques. In [21] we studied general elimination strategies, where one strategy which we called Neville elimination proved to be well suited to work with some special classes of matrices, in particular totally positive matrices (that are matrices with all subdeterminants nonnegative).…”

On Factorizations of Totally Positive Matrices

“…In some papers by M. Gasca and G. Mühlbach ( [13] for example) on the connection between interpolation formulas and elimination techniques it became clear that what they called Neville elimination had special interest for TP matrices. It is a procedure to make zeros in a column of a matrix by adding to each row an appropriate multiple of the precedent one and had been already used in some of the first papers on TP matrices [33].…”