Parallel QR Decomposition Research Articles

A blocked QR-decomposition for the parallel symmetric eigenvalue problem

• New parallel algorithm for the QR-decomposition of tall and skinny matrices (based on CholeskyQR). • Reduced synchronization requirements, compared to the classic Householder QR-decomposition. • Adaptive blocking during Householder vector generation, to guarantee numerical stability. • Considerable speedups were achieved on a BlueGene/P and Power6 system. In this paper we present a new stable algorithm for the parallel QR-decomposition of “tall and skinny” matrices. The algorithm has been developed for the dense symmetric eigensolver ELPA, where the QR-decomposition of tall and skinny matrices represents an important substep. Our new approach is based on the fast but unstable CholeskyQR algorithm (Stathopoulos and Wu, 2002) [1]. We show the stability of our new algorithm and provide promising results of our MPI-based implementation on a BlueGene/P and a Power6 system.

Parallel High Throughput Soft-Output Sphere Decoding Algorithm

Multiple-Input-Multiple-Output communication systems demand fast sphere decoding with high performance. To speed up the computation, we propose a scheme with multiple fixed complexity sphere decoders to construct a parallel soft-output fixed complexity sphere decoder (PFSD). The proposed decoder is highly parallel and has performance comparable to soft-output list fixed complexity sphere decoder (LFSD) and K-best sphere decoder. In addition, we propose a parallel QR decomposition algorithm to lower the preprocessing overhead, and a low complexity LLR algorithm to allow parallel update of LLR values. We demonstrate that the PFSD algorithm can increase the throughput and reduce bit error rate of a soft-output solution in a 4?×?4 16-QAM system, and has superior performance compared to other soft decoders with comparable throughput and computation complexity. The PFSD algorithm has been mapped onto Xilinx XC4VLX160 FPGA. The resulting PFSD decoder can achieve up to 75 Mbps throughput for 4?×?4 64-QAM configuration at 100MHz with low control overhead.

Load balanced parallel QR decomposition on shared memory multiprocessors

This paper introduces a new parallel QR decomposition algorithm with two key advantages over previous techniques. It is specifically designed to achieve high parallel efficiency on shared memory parallel computers with a modest number of processors, and the novel load balancing method described here considers total computational work as opposed to just balancing Givens rotations. This results in expected efficiencies which approach optimal as problem size grows relative to number of processors. The hybrid nature of the algorithm seeks to maximize computation between communication and synchronization. Implementation results on shared memory multiprocessors track expected performance well up to 12 processors, and initial performance results on a distributed shared memory machine are presented.

Multifrontal QR Factorization in a Multiprocessor Environment

We describe the design and implementation of a parallel QR decomposition algorithm for a large sparse matrix A. The algorithm is based on the multifrontal approach and makes use of Householder transformations. The tasks are distributed among processors according to an assembly tree which is built from the symbolic factorization of the matrix ATA. We first address uniprocessor issues and then discuss the multiprocessor implementation of the method. We consider the parallelization of both the factorization phase and the solve phase. We use relaxation of the sparsity structure of both the original matrix and the frontal matrices to improve the performance. We show that, in this case, the use of Level 3 BLAS can lead to very significant gains in performance. We use the eight processor Alliant˜FX/80 at CERFACS to illustrate our discussion.

A lower bound on the computational complexity of the QR decomposition on a shared memory SIMD computer

The time complexity of the Greedy algorithm is investigated. This algorithm provides the parallel QR decomposition of a dense rectangular matrix, through Givens rotations, and it is the optimal procedure within the class of parallel algorithms. The study is carried out analyzing the dynamical evolution of the Greedy automation that we introduce. The known lower bounds for the number of steps required by the Greedy algorithm, for the square matrix case, are improved.

Complexity of parallel QR factorization

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 2 N , when an unlimited number of processors is available.

Parallel QR decomposition of a rectangular matrix

We show that the greedy algorithm introduced in [1] and [5] to perform the parallel QR decomposition of a dense rectangular matrix of sizem×n is optimal. Then we assume thatm/n2 tends to zero asm andn go to infinity, and prove that the complexity of such a decomposition is asymptotically2n, when an unlimited number of processors is available.