A BLOCK ORTHOGONALIZATION PROCEDURE WITH CONSTANT SYNCHRONIZATION REQUIREMENTS

Abstract

We propose an alternative orthonormalization method that computes the orthonormal basis from the right singular vectors of a matrix. Its advantage are: a) all operations are matrix-matrix multiplications and thus cache-efficient, b) only one synchronization point is required in parallel implementations, c) could be more stable than Gram-Schmidt. In addition, we consider the problem of incremental orthonormalization where a block of vectors is orthonormalized against a previously orthonormal set of vectors and among itself. We solve this problem by alternating iteratively between a phase of Gram-Schmidt and a phase of the new method. We provide error analysis and use it to derive bounds on how accurately the two successive orthonormalization phases should be performed to minimize total work performed. Our experiments confirm the favorable numerical behavior of the new method and its effectiveness on modern parallel computers.