Complexity of the parallel Givens factorization on shared memory architectures

Abstract

We study the complexity of the parallel Givens factorization of a square matrix of size n on shared memory multicomputers with p processors. We show how to construct an optimal algorithm using a greedy technique. We deduce that the time complexity is equal to: $$T_{opt} (p) = \frac{{n^2 }}{{2p}} + p + o(n) for 1 \leqslant p \leqslant \frac{n}{{2 + \sqrt 2 }}$$ and that the minimum number of processors in order to compute the Givens factorization in optimal time Topt is equal to Popt=n/2+√2.These results complete previous analysis presented in the case where the number of processors is unlimited.KeywordsParallel linear algebracomplexity of parallel algorithmsorthogonal factorizationGivens rotations