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.

Parallel VLSI algorithm for stable inversion of dense matrices

Abstract: A fast, efficient, parallel algorithm for stable matrix inversion based on Givens plane rotations is described. The algorithm is implemented on VLSI systolic architecture that is capable of inverting any n*n nonsingular dense matrix in 5n units of time, including I/O time. The array architecture consists of n/sup 2/+n processing elements (PEs) arranged as a cascade of two triangular arrays of (n/sup 2/+n)/2 PEs each. The parallel algorithm involves the following processes: QR-decomposition by the Givens rotations technique; inversion of the upper triangular matrix R and multiplication of R/sup -1/ by Q. All three components of the algorithm are maximally overlapped with no need for intermediate I/O or global communications. A novel technique for concurrently inverting distinct matrices on the same array has also been suggested. This technique, based on the use of two levels of pipelining, significantly improves the array throughput. The execution speed of the algorithm matches that of the fastest systolic implementation of matrix inversion using Gaussian elimination without pivoting (which is unstable in most cases) reported to date.< >

Efficient denoising algorithms for large experimental datasets and their applications in Fourier transform ion cyclotron resonance mass spectrometry

Abstract: Modern scientific research produces datasets of increasing size and complexity that require dedicated numerical methods to be processed. In many cases, the analysis of spectroscopic data involves the denoising of raw data before any further processing. Current efficient denoising algorithms require the singular value decomposition of a matrix with a size that scales up as the square of the data length, preventing their use on very large datasets. Taking advantage of recent progress on random projection and probabilistic algorithms, we developed a simple and efficient method for the denoising of very large datasets. Based on the QR decomposition of a matrix randomly sampled from the data, this approach allows a gain of nearly three orders of magnitude in processing time compared with classical singular value decomposition denoising. This procedure, called urQRd (uncoiled random QR denoising), strongly reduces the computer memory footprint and allows the denoising algorithm to be applied to virtually unlimited data size. The efficiency of these numerical tools is demonstrated on experimental data from high-resolution broadband Fourier transform ion cyclotron resonance mass spectrometry, which has applications in proteomics and metabolomics. We show that robust denoising is achieved in 2D spectra whose interpretation is severely impaired by scintillation noise. These denoising procedures can be adapted to many other data analysis domains where the size and/or the processing time are crucial.

A Parallel Implementation of the Nonsymmetric QR Algorithm for Distributed Memory Architectures

Abstract: One approach to solving the nonsymmetric eigenvalue problem in parallel is to parallelize the QR algorithm. Not long ago, this was widely considered to be a hopeless task. Recent efforts have led to significant advances, although the methods proposed up to now have suffered from scalability problems. This paper discusses an approach to parallelizing the QR algorithm that greatly improves scalability. A theoretical analysis indicates that the algorithm is ultimately not scalable, but the nonscalability does not become evident until the matrix dimension is enormous. Experiments on the Intel Paragon system, the IBM SP2 supercomputer, the SGI Origin 2000, and the Intel ASCI Option Red supercomputer are reported.

Systolic Networks for Orthogonal Decompositions

Abstract: An orthogonally connected systolic array, consisting of a few types of simple processors, is constructed to perform the $QR$ decomposition of a matrix. Application is made to solution of linear systems and linear least squares problems as well as $QL$ and $LQ$ factorizations. For matrices A of bandwidth w the decomposition network requires less than $w^2 $ processors, independent of the order n of A. In terms of the operation time of the slowest processor, computation time varies between $2n$ and $4n$ subject to the number of codiagonals.