QR decomposition on GPUs

Abstract

QR decomposition is a computationally intensive linear algebra operation that factors a matrix A into the product of a unitary matrix Q and upper triangular matrix R. Adaptive systems commonly employ QR decomposition to solve overdetermined least squares problems. Performance of QR decomposition is typically the crucial factor limiting problem sizes.Graphics Processing Units (GPUs) are high-performance processors capable of executing hundreds of floating point operations in parallel. As commodity accelerators for 3D graphics, GPUs offer tremendous computational performance at relatively low costs. While GPUs are favorable to applications with much inherent parallelism requiring coarse-grain synchronization between processors, methods for efficiently utilizing GPUs for algorithms computing QR decomposition remain elusive.In this paper 1 , we discuss the architectural characteristics of GPUs and explain how a high-performance implementation of QR decomposition may be implemented. We provide detailed performance analysis of the resulting implementation for real-valued matrices and offer recommendations for achieving high performance to future developers of dense linear algebra procedures for GPUs. Our implementation sustains 143 GFLOP/s, and we believe this is the fastest announced QR implementation executing entirely on the GPU.

Transient Fault Resilient QR Factorization on GPUs

“…This algorithm has O(n 3 ) complexity. For more details, we refer the interested reader to the comprehensive overview by Kerr et al [13]. Note that the block Householder algorithm utilizes the house() function, shown in Algorithm 2.…”

Floating Point Architecture Extensions for Optimized Matrix Factorization

“…The details of multi-core, multi-GPU QR factorization scheduling are discussed in [3]. A solution for QR factorization that can be entirely run on the GPU is presented in [7]. For LU factorization on GPUs, a technique to reduce matrix decomposition and row operations to a series of rasterization problems is used [8].…”

Algorithm/Architecture Codesign of Low Power and High Performance Linear Algebra Compute Fabrics

“…A solution for QR factorization that can be entirely run on the GPU is presented in [71]. For LU factorization on GPUs, a technique to reduce matrix decomposition and row operations to a series of rasterization problems is used [44].…”