QR decomposition

About: QR decomposition is a research topic. Over the lifetime, 3504 publications have been published within this topic receiving 100599 citations. The topic is also known as: QR factorization.

A multilevel implementation of the QR compression for method of moments

Abstract: We present a multilevel version of a previously reported compression technique for the method of moments (Poirier, J.-R. et al., ibid., vol.46, p.1175-6, 1998). This method is based on QR compression of the impedance matrix off-diagonal blocks. The block structure of the matrix is built up to optimize the compression rate. It results from the study that the compression threshold is the unique adjustment parameter, which makes this method very easy to handle. Some results are presented, which stress the improvement in memory storage and computation time obtained through this approach with respect to the mono-level algorithm.

A practical application of kernel-based fuzzy discriminant analysis

Abstract: A novel method for feature extraction and recognition called Kernel Fuzzy Discriminant Analysis KFDA is proposed in this paper to deal with recognition problems, e.g., for images. The KFDA method is obtained by combining the advantages of fuzzy methods and a kernel trick. Based on the orthogonal-triangular decomposition of a matrix and Singular Value Decomposition SVD, two different variants, KFDA/QR and KFDA/SVD, of KFDA are obtained. In the proposed method, the membership degree is incorporated into the definition of between-class and within-class scatter matrices to get fuzzy between-class and within-class scatter matrices. The membership degree is obtained by combining the measures of features of samples data. In addition, the effects of employing different measures is investigated from a pure mathematical point of view, and the t-test statistical method is used for comparing the robustness of the learning algorithm. Experimental results on ORL and FERET face databases show that KFDA/QR and KFDA/SVD are more effective and feasible than Fuzzy Discriminant Analysis FDA and Kernel Discriminant Analysis KDA in terms of the mean correct recognition rate.

Accurate downdating of a modified Gram-Schmidt QR decomposition

Abstract: A new algorithm for downdating a QR decomposition is presented. We show that, when the columns in the Q factor from the Modified Gram-Schmidt QR decomposition of a matrixX are exactly orthonormal, the Gram-Schmidt downdating algorithm for the QR decomposition ofX is equivalent to downdating the full Householder QR decomposition of the matrixX augmented by ann ×n zero matrix on top. Using this relation, we derive an algorithm that improves the Gram-Schmidt downdating algorithm when the columns in the Q factor are not orthonormal. Numerical test results show that the new algorithm produces far more accurate results than the Gram-Schmidt downdating algorithm for certain ill-conditioned problems.

Efficient QR Updating Factorization for Sensorless Synchronous Motor Drive Based on High Frequency Voltage Injection

Abstract: The conventional methods for estimating the rotor position of permanent magnet synchronous machines, at low speed range and characterized by rotor saliency, rely on high frequency voltage injection in the stator windings. Ordinarily, the rotor position estimation is achieved through the demodulation of the high frequency current response. In this article, an alternative method is presented for detecting rotor position from the rotating high frequency injection current response. The proposed ellipse fitting technique is not affected by signal processing delay effects and it requires the tuning of only one parameter, called forgetting factor, making the studied method suitable for industrial application thanks to its minimal setup effort. The inverse problem related to the ellipse fitting is solved implementing a QR recursive least squares algorithm. Efficient updating QR factorization has been adopted because of its features in terms of numerical stability and required limited computational effort. The proposed sensorless control scheme is validated by means of many experiments.

Randomized Gram-Schmidt process with application to GMRES.

Abstract: A randomized Gram-Schmidt algorithm is developed for orthonormalization of high-dimensional vectors or QR factorization. The proposed process can be less computationally expensive than the classical Gram-Schmidt process while being at least as numerically stable as the modified Gram-Schmidt process. Our approach is based on random sketching, which is a dimension reduction technique consisting in estimation of inner products of high-dimensional vectors by inner products of their small efficiently-computable random projections, so-called sketches. This allows to perform the projection step in Gram-Schmidt process on sketches rather than high-dimensional vectors with a minor computational cost. This also provides an ability to efficiently certify the output. The proposed Gram-Schmidt algorithm can provide computational cost reduction in any architecture. The benefit of random sketching can be amplified by exploiting multi-precision arithmetic. We provide stability analysis for multi-precision model with coarse unit roundoff for standard high-dimensional operations. Numerical stability is proven for the unit roundoff independent of the (high) dimension of the problem. The proposed Gram-Schmidt process can be applied to Arnoldi iteration and result in new Krylov subspace methods for solving high-dimensional systems of equations or eigenvalue problems. Among them we chose randomized GMRES method as a practical application of the methodology.