Updating and downdating of orthogonal polynomials with data fitting applications

New methods for updating and downdating least squares polynomial fits to discrete data are derived and assessed using polynomials orthogonal on all the data points being used. Rather than fixing on one basis throughout, the methods adaptively update and downdate both the least squares fit and the polynomial basis. This is achieved by performing similarity transformations on the tridiagonal Jacobi matrices representing the basis. Although downdating is potentially unstable, experimental results show that the methods give satisfactory results for low degree fits. Details of new algorithms implementing the methods are given, the most economical of which needs $14n + O ( 1 )$ flops and $2n$ square roots to update a fit of order $n$.