Least squares problems involving generalized Kronecker products and application to bivariate polynomial regression

A method for solving least squares problems (A ⊗ Bi)x = b whose coefficient matrices have generalized Kronecker product structure is presented. It is based on the exploitation of the block structure of the Moore-Penrose inverse and the reflexive minimum norm g-inverse of the coefficient matrix, and on the QR method for solving least squares problems. Firstly, the general case where A is a rectangular matrix is considered, and then the special case where A is square is analyzed. This special case is applied to the problem of bivariate polynomial regression, in which the involved matrices are structured matrices (Vandermonde or Bernstein-Vandermonde matrices). In this context, the advantage of using the Bernstein basis instead of the monomial basis is shown. Numerical experiments illustrating the good behavior of the proposed algorithm are included.Ministerio de Economía y Competitivida

Accurate computations with collocation matrices of the Lupaş-type (p,q)-analogue of the Bernstein basis

A fast and accurate algorithm to compute the bidiagonal decomposition of collocation matrices of the Lupaş-type (p,q)-analogue of the Bernstein basis is presented. The error analysis of the algorithm and the perturbation theory for the bidiagonal decomposition are also included. Starting from this bidiagonal decomposition, the accurate and efficient solution of several linear algebra problems involving these matrices is addressed: linear system solving, eigenvalue and singular value computation, and computation of the inverse and the Moore-Penrose inverse. The numerical experiments carried out show the good behaviour of the algorithm.Agencia Estatal de Investigació

Error analysis, perturbation theory and applications of the bidiagonal decomposition of rectangular totally-positive h-Bernstein-Vandermonde matrices

A fast and accurate algorithm to compute the bidiagonal decomposition of rectangular totally positive h-Bernstein-Vandermonde matrices is presented. The error analysis of the algorithm and the perturbation theory for the bidiagonal decomposition of totally positive h-Bernstein-Vandermonde matrices are addressed. The computation of this bidiagonal decomposition is used as the first step for the accurate and efficient computation of the singular values of rectangular totally positive h-Bernstein-Vandermonde matrices and for solving least squares problems whose coefficient matrices are such matrices.Agencia Estatal de Investigació

Extracción de información de observaciones hiperespectrales en teledetección

La aparición reciente de datos hiperespectrales en Teledetección supone disponer de una gran cantidad de información que hay que procesar y analizar. Aunque las técnicas de análisis existentes para datos de pocas dimensiones pueden ser usadas para analizar datos de gran dimensión, surgen algunos problemas en el análisis hiperespectral que no existían en el análisis de datos multiespectrales. Aquí exponemos ventajas e inconvenientes de distintos métodos paira obtener la máxima información de estos nuevos sistemas hiperespectrales