43 research outputs found
Bidiagonal factorizations with some parameters equal to zero
AbstractMotivated by the results of Fiedler and Markham [2], we provide necessary and sufficient conditions for a matrix to have a bidiagonal factorization with some of the parameters of the bidiagonal factors equal to zero
Accurate computations with Wronskian matrices
In this paper we provide algorithms for computing the bidiagonal decomposition of the Wronskian matrices of the monomial basis of polynomials and of the basis of exponential polynomials. It is also shown that these algorithms can be used to perform accurately some algebraic computations with these Wronskian matrices, such as the calculation of their inverses, their eigenvalues or their singular values and the solutions of some linear systems. Numerical experiments illustrate the results
Total positivity and least squares problems in the Lagrange basis
The problem of polynomial least squares fitting in the standard Lagrange
basis is addressed in this work. Although the matrices involved in the
corresponding overdetermined linear systems are not totally positive,
rectangular totally positive Lagrange-Vandermonde matrices are used to take
advantage of total positivity in the construction of accurate algorithms to
solve the considered problem. In particular, a fast and accurate algorithm to
compute the bidiagonal decomposition of such rectangular totally positive
matrices is crucial to solve the problem. This algorithm also allows the
accurate computation of the Moore-Penrose inverse and the projection matrix of
the collocation matrices involved in these problems. Numerical experiments
showing the good behaviour of the proposed algorithms are included
Accurate computations with Wronskian matrices of Bessel and Laguerre polynomials
This paper provides an accurate method to obtain the bidiagonal factorization of Wronskian matrices of Bessel polynomials and of Laguerre polynomials. This method can be used to compute with high relative accuracy their singular values, the inverse of these matrices, as well as the solution of some related systems of linear equations. Numerical examples illustrating the theoretical results are included. © 2022 The Author
Depth of almost strictly sign regular matrices
The concept of depth of an almost strictly sign regular matrix is introduced and used to simplify some algorithmic characterizations of these matrices