Accurate bidiagonal decomposition of collocation matrices of weighted φ-transformed systems

Given a system of functions, we introduce the concept of weighted phi-transformed system, which will include a very large class of useful representations in Statistics and Computer Aided Geometric Design. An accurate bidiagonal decomposition of the collocation matrices of these systems is obtained. This decomposition is used to present computational methods with high relative accuracy for solving algebraic problems with collocation matrices of weighted phi-transformed systems such as the computation of eigenvalues, singular values, and the solution of some linear systems. Numerical examples illustrate the accuracy of the performed computations.