Optimal algorithms for parallel Givens factorization on a coarse-grained PRAM

Abstract

We study the complexity of the parallel Givens factorization of a square matrix of size n on a shared memory architecture composed with p identical processors (coarse grained EREW PRAM). We show how to construct an asymptotically optimal algorithm. We deduce that the time complexity is equal to:and that the minimum number of processors in order to compute the Givens factorization in asymptotically optimal time (2n + o(n)) is equal to pOPt = n/(2These results complete previous analysis presented in the case where the number of processors is unlimited.

Parallel Algorithms for Linear Models

“…This particular feature underpins the development of parallel Givens algorithms for solving a range of matrix factorization problems [29,30,69,94,95,102,103,129].…”

Orthogonal Factorization and Linear Least Squares Problems

“…i,i+1 to annihilate the entry (i + 1, j) ; 11: end 12: end Other orderings for parallel Givens reduction are given in [7][8][9], with the ordering presented here judged as being asymptotically optimal [10].…”

Parallel Givens Sequences for Solving the General Linear Model on a Erew Pram∗

“…Similarly, the Givens rotation that annihilates the element of C at position (i, i -1) when applied from the right of C is defined as parallel Givens algorithms for solving a range of matrix factorization problems [3,4,9,[19][20][21][22]281. The complexity analysis of these algorithms is often based on the assumption that a single time unit is needed for the simultaneous application of disjoint Givens rotations on any matrix conformal with the transformation.…”

“…The unrealistic unlimited parallelism assumption is restricted so that, in a single time unit, the number of Givens rotations applied simultaneously is limited and the time to perform the transformations depends on the length of the vectors affected by the compound rotations. An EREW (Exclusive ReadExclusive Write) PRAM (Parallel Random Access Machine) computational model will be used in the development and complexity analysis of the Givens algorithms [3]. A single time unit is defined to be the time required to execute the operation of applying a Givens rotation to a 2-element vector.…”