Householder's method

About: Householder's method is a research topic. Over the lifetime, 122 publications have been published within this topic receiving 3476 citations.

Orthogonal least squares methods and their application to non-linear system identification

Abstract: Identification algorithms based on the well-known linear least squares methods of gaussian elimination, Cholesky decomposition, classical Gram-Schmidt, modified Gram-Schmidt, Householder transformation, Givens method, and singular value decomposition are reviewed. The classical Gram-Schmidt, modified Gram-Schmidt, and Householder transformation algorithms are then extended to combine structure determination, or which terms to include in the model, and parameter estimation in a very simple and efficient manner for a class of multivariate discrete-time non-linear stochastic systems which are linear in the parameters.

The WY representation for products of householder matrices

Abstract: A new way to represent products of Householder matrices is given that makes a typical Householder matrix algorithm rich in matrix-matrix multiplication. This is very desirable in that matrix-matrix...

Hyperbolic householder transformations

Abstract: A class of transformation matrices, analogous to the Householder matrices, is developed, with a nonorthogonal property designed to permit the efficient deletion of data from least-squares problems. These matrices, which we term hyperbolic Householder, are shown to effect deletion, or simultaneous addition and deletion, of data with much less sensitivity to rounding errors than for techniques based on normal equations. When the addition/deletion sets are large, this numerical robustness is obtained at the expense of only a modest increase in computations, and when only a relatively small fraction of the data set is modified, there is a decrease in required computations. Two applications to signal processing problems are considered. First, these transformations are used to obtain a square root algorithm for windowed recursive least-squares filtering. Second, the transformations are employed to implement the rejection of spurious data from the weight vector estimator in an adaptive array.

Estimation of the arrival times of seismic waves by multivariate time series model

Abstract: A computationally efficient procedure was developed for the fitting of many multivariate locally stationary autoregressive models. The details of the Householder method for fitting multivariate autoregressive model and multivariate locally stationary autoregressive model (MLSAR model) are shown. The proposed procedure is quite efficient in both accuracy and computation. The amount of computation is bounded by a multiple of Nm 2 with N being the data length and m the highest model order, and does not depend on the number of models checked. This facilitates the precise estimation of the change point of the AR model. Based on the AICs' of the fitted MLSAR models and Akaike's definition of the likelihood of the models, a method of evaluating the posterior distribution of the change point of the AR model is also presented. The proposed procedure is, in particular, useful for the estimation of the arrival time of the S wave of a microearthquake. To illustrate the usefulness of the proposed procedure, the seismograms of the foreshocks of the 1982 Urakawa-Oki Earthquake were analyzed. These data sets have been registered to AISM Data Library and the readers of this Journal can access to them by the method described in this issue.

A Block Orthogonalization Procedure with Constant Synchronization Requirements

Abstract: First, we consider the problem of orthonormalizing skinny (long) matrices. We propose an alternative orthonormalization method that computes the orthonormal basis from the right singular vectors of a matrix. Its advantages are that (a) all operations are matrix-matrix multiplications and thus cache efficient, (b) only one synchronization point is required in parallel implementations, and (c) it is typically more stable than classical Gram--Schmidt (GS). Second, we consider the problem of orthonormalizing a block of vectors against a previously orthonormal set of vectors and among itself. We solve this problem by alternating iteratively between a phase of GS and a phase of the new method. We provide error analysis and use it to derive bounds on how accurately the two successive orthonormalization phases should be performed to minimize total work performed. Our experiments confirm the favorable numerical behavior of the new method and its effectiveness on modern parallel computers.