On Factorizations of Totally Positive Matrices

Abstract. Different approaches to the decomposition of a nonsingular totally positive matrix as a product of bidiagonal matrices are studied. Special attention is paid to the interpretation of the factorization in terms of the Neville elimination process of the matrix and in terms of corner cutting algorithms of Computer Aided Geometric Design. Conditions of uniqueness for the decomposition are also given. Totally positive matrices (TP matrices in the sequel) are real, nonnegative matrices whose all minors are nonnegative. They have a long history and many applications (see the paper by Allan Pinkus in this volume for the early history and motivations) and have been studied mainly by researchers of thos

Biorthogonal Wavelets and Multigrid

We will be concerned with the solution of an elliptic boundary value problem in one dimension with polynomial coefficients. In a Galerkin approach, we employ biorthogonal wavelets adapted to a differential operator with constant coefficients, and use the refinement equations to set up the system of linear equations with exact entries (up to round-off). For the solution of the linear equation, we construct a biorthogonal two-grid method with intergrid operators stemming from wavelet-type operators adapted to the problem. Key words: Adapted biorthogonal wavelets, boundary value problems, refinement equations, two-grid (multi-grid) methods. AMS subject classification: 65L10, 65L60, 65F10, 41A30. 1 Introduction When solving elliptic boundary value problems by means of a Galerkin approach, the trial functions for the approximation spaces are usually chosen with compact support such that the resulting stiffness matrices are as sparse as possible. But even then, the resulting system of equ..