The Inverse QR Decomposition in Order-recursive Calculation of Least Squares Coefficients

In this paper we examine the properties of QR and inverse QR factorizations in the general linear least squares (LS) problem. By exploiting a straightforward geometric interpretation of the factorization, an efficient algorithm is derived that provides, order recursively, the LS coefficient vector, projection error vector, and residual error energy (i.e. the sum of the squares of the elements of the error vector) for all of the LS problems as the order varies from one to n, n being a prespecified maximum order. Using existing algorithms for time updating the inverse QR factorization, the method applies to the time recursive situation also. Given only R-l and the last row of Q in the inverse QR factorization of the data covariance matrix, all order updates of the LS coefficient vectors and residual error energies are carried out. Application to multichannel adaptive LS filtering is presented.