A general orthogonalization technique with applications to time series analysis and signal processing

Abstract

A new orthogonalization technique is presented for computing the QR factorization of a general n × p n \times p matrix of full rank p ( n ⩾ p ) p\,(n \geqslant p) . The method is based on the use of projections to solve increasingly larger subproblems recursively and has an O ( n p 2 ) O(n{p^2}) operation count for general matrices. The technique is readily adaptable to solving linear least-squares problems. If the initial matrix has a circulant structure the algorithm simplifies significantly and gives the so-called lattice algorithm for solving linear prediction problems. From this point of view it is seen that the lattice algorithm is really an efficient way of solving specially structured least-squares problems by orthogonalization as opposed to solving the normal equations by fast Toeplitz algorithms.