  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.
