Abstract: A new orthogonalization technique is presented for computing the QR factorization of a general n X p matrix of full rank p (n 2 p). The method is based on the use of projections to solve increasingly larger subproblems recursively and has an O(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.

Abstract: A novel approach to nonlinear filtering with minimum mean square error criterion is presented. This method considers the class of nonlinear filters with Volterra series structures under the assumption that filter inputs are Gaussian, and a relatively simple solution results which is directly applicable in many practical situations. Moreover, two simple parameter adaption algorithms for the second order Volterra filter are presented and it is shown that their convergence speeds depend on the squared ratio of maximum to minimum eigenvalues of the input autocovariance matrix. Finally, the lattice orthogonalization of filter input is considered for faster convergence.

Abstract: A cardiogoniometry or vector cardiography system wherein signals directly derived from the bioelectrical field are not directly processed as orthogonal data but are instead especially orthogonalized in an analog computing network. Orthogonalization is based on a derivation electrode configuration space with sloping sagittal and frontal planes. The orthogonalized signals are processed in a cardiogoniometer and are also jointly recorded on a commercial electrocardiograph in parallel thereto. All the represented data which can be used for diagnosis purposes are referenced to a biological zero line, which differs from the electrical neutral point. The cardiogoniometer permits a vectorial real time measurement on the patient.

Abstract: How ill-conditioned must a matrix S be if it (block) diagonalizes a given matrix T, i.e. if S -1 TS is block diagonal? The answer depends on how the diagonal blocks partition T''s spectrum; the condition number of S is bounded below by a function of the norms of the projection matrices determined by the partitioning. In the case of two diagonal blocks we compute an S which attains this lower bound, and we describe almost best conditioned S''s for dividing T into more blocks. We apply this result to bound the error in an algorithms to compute analytic functions of matrices, for instance exp(T). Our techniques also produce bounds for submatrices that appear in the square-root-free Choleskt and in the Gram-Schmidt orthogonalization algorithms.