A General Orthogonalization Technique With Applications to Time Series Analysis

A new orthogonalization technique is presented for computing the QR factorization of a general n X p matrix of full rank p (n > p). The method is based on the use of projections to solve increasingly larger subproblems recursively and has an 0(np2) 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.