Order-recursive factorization of the pseudoinverse of a covariance matrix

An order-recursive method for the computation of a square root factor of a pseudoinverse of a covariance matrix is given. For an N×N covariance matrix and a nested sequence of N submatrices, the pseudoinverse factors are computed for all N submatrices in order N³ operations, whereas direct computation requires order N⁴. Computational algorithms are given in terms of a generalized SVD involving explicit rank decisions concerning zero variance and unity correlation. Applications to estimation and parameter identification are discussed. Numerical accuracy is illustrated by an example. An operation count is given for serial computing as well as for partitioning of the algorithm in parallel computing.