In this paper, we present a new parallel accurate algorithm called PAccSumK for computing summation of floating-point numbers. It is based on AccSumK algorithm. In the experiment, for the summation problems with large condition numbers, our algorithm outperforms in term of accuracy and computing time the PSumK algorithm. The reason is that our algorithm is based on a more accurate algorithm called AccSumK algorithm compared to the SumL algorithm used in PSumK. The proposed parallel algorithms in this paper is designed to compute a result as if computed internally in K-fold the working precision. Numerical results are presented showing the performance and the accuracy of our new parallel algorithm for calculating summation.
Krylov subspace algorithms are important methods for solving linear systems. In order to solve large-scale linear systems and speed up the solution of linear systems, one has to use parallelism techniques. However, parallelism often enlarge the non-associativity of floating-point operations. This can lead to non-reproducibility of the computations. This paper compares the performance of the parallel preconditioned BiCGSTAB algorithm implemented with two different libraries (ExBLAS and ReproBLAS) that can ensure reproducibility of the computations. To address the effect of the compiler, we explicitly utilize the fma instructions. Finally, numerical experiments show that the BiCGSTAB algorithms based on the two BLAS implementations are reproducible, the BiCGSTAB algorithm based on ExBLAS is more accurate but more time-consuming, and the BiCGSTAB algorithm based on ReproBLAS is relatively less accurate but less expensive.
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