Wyniki wyszukiwania - BazTech

1

The parallel tiled WZ factorization algorithm for multicore architectures

Bylina Beata, Bylina Jarosław

International Journal of Applied Mathematics and Computer Science

|

2019

|

Vol. 29, no. 2

407--419

EN

The aim of this paper is to investigate dense linear algebra algorithms on shared memory multicore architectures. The design and implementation of a parallel tiled WZ factorization algorithm which can fully exploit such architectures are presented. Three parallel implementations of the algorithm are studied. The first one relies only on exploiting multithreaded BLAS (basic linear algebra subprograms) operations. The second implementation, except for BLAS operations, employs the OpenMP standard to use the loop-level parallelism. The third implementation, except for BLAS operations, employs the OpenMP task directive with the depend clause. We report the computational performance and the speedup of the parallel tiled WZ factorization algorithm on shared memory multicore architectures for dense square diagonally dominant matrices. Then we compare our parallel implementations with the respective LU factorization from a vendor implemented LAPACK library. We also analyze the numerical accuracy. Two of our implementations can be achieved with near maximal theoretical speedup implied by Amdahl’s law.

2

Influence of preconditioning and blocking on accuracy in solving Markovian models

Bylina B., Bylina J.

International Journal of Applied Mathematics and Computer Science

|

2009

|

Vol. 19, no 2

207-217

EN

The article considers the effectiveness of various methods used to solve systems of linear equations (which emerge while modeling computer networks and systems with Markov chains) and the practical influence of the methods applied on accuracy. The paper considers some hybrids of both direct and iterative methods. Two varieties of the Gauss elimination will be considered as an example of direct methods: the LU factorization method and the WZ factorization method. The Gauss-Seidel iterative method will be discussed. The paper also shows preconditioning (with the use of incomplete Gauss elimination) and dividing the matrix into blocks where blocks are solved applying direct methods. The motivation for such hybrids is a very high condition number (which is bad) for coefficient matrices occuring in Markov chains and, thus, slow convergence of traditional iterative methods. Also, the blocking, preconditioning and merging of both are analysed. The paper presents the impact of linked methods on both the time and accuracy of finding vector probability. The results of an experiment are given for two groups of matrices: those derived from some very abstract Markovian models, and those from a general 2D Markov chain.

3

Rozwiązywanie układów równań metodą rozkładu WZ z wykorzystaniem bibliotek BLAS1 i BLAS2

Bylina B., Bylina J.

Studia Informatica

|

2004

|

Vol. 25, nr 1

57-67

PL

W artykule zaprezentowano strategię wyboru elementu podstawowego, efektywny blokowo-punktowy algorytm rozwiązywania układów równań liniowych metodą rozkładu WZ macierzy na czynniki z wyborem elementu podstawowego oraz szczegóły jego implementacji z wykorzystaniem bibliotek BLAS. Przedstawiony algorytm rozkładu WZ jest szybszy niż klasyczny rozkład sekwencyjny - również niż ten z włączoną optymalizacją kompilatora.

EN

In this article we want to present a strategy of pivoting, an efficient matrix-vector algorithm for solving linear systems by WZ matrix factorization with pivoting and details of its implementation with usage of BLAS libraries. The presented algorithm is faster than the seąuential one.