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 Class of Intrinsic Schemes for Orthogonal Integration

Abstract: Numerical integration of ODEs on the orthogonal Stiefel manifold is considered. Points on this manifold are represented as n × k matrices with orthonormal columns, of particular interest is the case when $n\gg k$. Mainly two requirements are imposed on the integration schemes. First, they should have arithmetic complexity of order nk2. Second, they should be intrinsic in the sense that they require only the ODE vector field to be defined on the Stiefel manifold, as opposed to, for instance, projection methods. The design of the methods makes use of retractions maps. Two algorithms are proposed, one where the retraction map is based on the QR decomposition of a matrix, and one where it is based on the polar decomposition. Numerical experiments show that the new methods are superior to standard Lie group methods with respect to arithmetic complexity, and may be more reliable than projection methods, owing to their intrinsic nature.

A Hierarchical Approach for Performance Analysis of ScaLAPACK-Based Routines Using the Distributed Linear Algebra Machine

Abstract: Performance models are important in the design and analysis of linear algebra software for scalable high performance computer systems. They can be used for estimation of the overhead in a parallel algorithm and measuring the impact of machine characteristics and block sizes on the execution time. We present an hierarchical approach for design of performance models for parallel algorithms in linear algebra based on a parallel machine model and the hierarchical structure of the ScaLAPACK library. This suggests three levels of performance models corresponding to existing ScaLAPACK routines. As a proof of the concept a performance model of the high level QR factorization routine pdgeqrf is presented. We also derive performance models of lower level ScaLAPACK building blocks such as pdgeqr2, pdlarft, pdlarfb, pdlarfg, pdlarf, pdnrm2, and pdscal, which are used in the high level model for pdgeqrf. Predicted performance results are compared to measurements on an Intel Paragon XP/S system. The accuracy of the top level model is over 90% for measured matrix and block sizes and different process grid configurations.

Computational Workloads for Commonly Used Signal Processing Kernels

Abstract: : In the course of designing or evaluating signal processing algorithms, one often must determine the computational workload needed to implement the algorithms on a digital computer. The floating-point operation (flop) counts for real versions of the most common signal processing kernels are well documented. However, the flop counts for kernels operating on complex inputs are not as readily found. This report collects the flop count expressions for both real and complex kernels and also presents brief outlines of the derivations for the flop count expressions. Specifically, the following computational kernels are addressed: (1) the dimensions of the two multiplicands (m x n and n x p) for the matrix-matrix multiplication; (2) the length of the vector n for the fast Fourier transform; (3) the size of the triangular system n for forward and back substitutions; (4) the dimensions of the input matrix m x n for the Householder QR decomposition, eigenvalue decomposition, and singular value decomposition.

Complexity of parallel QR factorization

Abstract: An optimal algorithm to perform the parallel QR decomposition of a dense matrix of size N is proposed. It is deduced that the complexity of such a decomposition is asymptotically 2N, when an unlimited number of processors is available.