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.

The torus-wrap mapping for dense matrix calculations on massively parallel computers

Abstract: Dense linear systems of equations are quite common in science and engineering, arising in boundary element methods, least squares problems, and other settings. Massively parallel computers will be necessary to solve the large systems required by scientists and engineers, and scalable parallel algorithms for the linear algebra applications must be devised for these machines. A critical step in these algorithms is the mapping of matrix elements to processors. In this paper, the use of the torus-wrap mapping in general dense matrix algorithms is studied from both theoretical and practical viewpoints. Under reasonable assumptions, it is proved that this assignment scheme leads to dense matrix algorithms that achieve (to within a constant factor) the lower bound on interprocessor communication. It is also shown that the torus-wrap mapping allows algorithms to exhibit less idle time, better load balancing, and less memory overhead than the more common row and column mappings. Finally, practical implementation i...

A Method for Recursive Least Squares Filtering Based Upon an Inverse QR Decomposition

Abstract: A new computationally efficient algorithm for re- cursive least squares filtering is derived, which is based upon an inverse QR decomposition. The method solves directly for the time-recursive least squares filter vector, while avoiding the highly serial backsubstitution step required in previously de- rived direct QR approaches. Furthermore, the method employs orthogonal rotation operations to recursively update the filter, and thus preserves the inherent stability properties of QR ap- proaches to recursive least squares filtering. The results of sim- ulations over extremely long data sets are also presented, which suggest stability of the new time-recursive algorithm. Finally, parallel implementation of the resulting method is briefly dis- cussed, and computational wavefronts are displayed.

VLSI Architecture for Matrix Inversion using Modified Gram-Schmidt based QR Decomposition

Abstract: Matrix inversion and triangularization problems are common to a wide variety of communication systems, signal processing applications and solution of a set of linear equations. Matrix inversion is a computationally intensive process and its hardware implementation based on fixed-point (FP) arithmetic is a challenging problem. This paper proposes a fully parallel VLSI architecture under fixed-precision for the inverse computation of a real square matrix using QR decomposition with modified Gram-Schmidt (MGS) orthogonalization. The MGS algorithm is stable and accurate to the integral multiples of machine precision under fixed-precision for a well-conditioned non-singular matrix. For typical matrices (4 times 4) found in MIMO communication systems, the proposed architecture was able to achieve a clock rate of 277 MHz with a latency of 18 time units and area of 72K gates using 0.18-mum CMOS technology. For a generic square matrix of order n, the latency required is 5n - 2 which is better than all previously known architectures. With the use of LUTs and log-domain computations, the total area has been reduced compared to architectures based on linear-domain computations

A fast and well-conditioned spectral method

Abstract: A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients. The method leads to matrices which are almost banded, and a numerical solver is presented that takes O(m^2n) operations, where m is the number of Chebyshev points needed to resolve the coefficients of the differential operator and n is the number of Chebyshev coefficients needed to resolve the solution to the differential equation. We prove stability of the method by relating it to a diagonally preconditioned system which has a bounded condition number, in a suitable norm. For Dirichlet boundary conditions, this implies stability in the standard 2-norm. An adaptive QR factorization is developed to efficiently solve the resulting linear system and automatically choose the optimal number of Chebyshev coefficients needed to represent the solution. The resulting algorithm can efficiently and reliably solve for solutions that require as many as a million unknowns.

Joint data QR-detection and Kalman estimation for OFDM time-varying Rayleigh channel complex gains

Abstract: This paper deals with the case of a high speed mobile receiver operating in an orthogonal-frequency-division-multiplexing (OFDM) communication system. Assuming the knowledge of delay-related information, we propose an iterative algorithm for joint multi-path Rayleigh channel complex gains estimation and data recovery in fast fading environments. Each complex gain time-variation, within one OFDM symbol, is approximated by a polynomial representation. Based on the Jakes process, an auto-regressive (AR) model of the polynomial coefficients dynamics is built, making it possible to employ the Kalman filter estimator for the polynomial coefficients. Hence, the channel matrix is easily computed, and the data symbol is estimated with free inter-sub-carrier-interference (ICI) thanks to the use of a QR-decomposition of the channel matrix. Our claims are supported by theoretical analysis and simulation results, which are obtained considering Jakes' channels with high Doppler spreads.