Numerical analysis of parallel implementation of the reorthogonalized ABS methods

Abstract

Solving systems of equations is a critical step in various computational tasks. We recently published a novel ABS-based reorthogonalization algorithm to compute the QR factorization. Experimental analysis on the Matlab 2015a platform revealed that this new ABS-based algorithm was able to more accurately calculate the rank of the coefficient matrix, the determination of the orthogonal bases and the QR factorization than the built-in rank or qr Matlab functions. However, the reorthogonalization process significantly increased the computation cost. Therefore, we tested a new approach to accelerate this algorithm by implementing it on different parallel platforms. The above mentioned ABS-based reorthogonalization algorithm was implemented using Matlab’s parallel computing toolbox and accelerated massive parallelism (AMP) runtime library. We have tested various matrices including Pascal, Vandermonde and randomly generated dense matrices. The performance of the parallel algorithm was determined by calculating the speed-up factor defined as the fold reduction of execution time compared to the sequential algorithm. For comparison, we also tested the effect of parallel implementation of the classical Gram–Schmidt algorithm incorporating a reorthogonalization step. The results show that the achieved speed-up is significant, and also the performance of this practical parallel algorithm increases as the number of equations grows. The results reveal that the reorthogonalized ABS algorithm is practical and efficient. This fact expands the practical usefulness of our algorithms.