requirement and introduces the necessity of loose synchronization in the parallel execution of an operation. This paper presents an algorithm for the QR factorization where the operations can be represented as a sequence of small tasks that operate on square blocks of data. These tasks can be dynamically

The purpose of this note is to re-introduce the generalized QR factorization with or without pivoting of two matrices A and B having the same numberofrows, and whenever B is square and nonsingular, the factorization implicitly gives the orthogonal factorization with or without pivoting of B,1 A

Abstract. This manuscript focuses on the development of a parallel QR-factorization of structured rank matrices, which can then be used for solving systems of equations. First, we will prove the existence of two types of Givens transformations, named rank decreasing and rank expanding Givens

Abstract. In this paper we present unblocked and blocked LAPACKstyle codes for the computation of the QR factorization of a banded matrix. The new routines are evaluated using highly-tuned multithreaded implementations of BLAS on two shared-memory multiprocessors, revealing the performance

For sparse QR factorization, finding a good column ordering of the matrix to be factorized, is essential. Both the amount of fill in the resulting factors, and the number of floating-point operations required by the factorization, are highly dependent on this ordering. A suitable column ordering

the rotation graph as a tool to devise elimination orderings in QR factorizations. Properties of this graph characterize sets of rotation planes that are sufficient (or sufficient under permutation) and identify rotation planes to add to complete a deficient set. We also devise a constructive way to determine

Hessenberg–Toeplitz matrices

SuiteSparseQR is a sparse QR factorization package based on the multifrontal method. Within each frontal matrix, LAPACK and the multithreaded BLAS enable the method to obtain high performance on multicore architectures. Parallelism across different frontal matrices is handled with Intel’s Threading

We present a parallel algorithm for the QR factorization with column pivoting of a sparse matrix by means of Givens rotations. Nonzero elements of the matrix M to be decomposed are stored in a onedimensional doubly linked list data structure. We will discuss a strategy to reduce fill-in in order

In [5, 6], we presented algorithm RGEQR3, a purely recursive formulation of the QR factorization. Using recursion leads us to a natural way to choose the k-way aggregating Householder transform of Schreiber and Van Loan [10]. RGEQR3 is a performance critical subroutine for the main (hybrid