On Using the Cholesky QR Method in the Full-Blocked One-Sided Jacobi Algorithm

Abstract

The one-sided Jacobi method is known as an alternative of the bi-diagonalization based singular value decomposition (SVD) algorithms like QR, divide-and-conquer and MRRR, because of its accuracy and comparable performance. There is an extension of the one-sided Jacobi method called “full-blocked” method, which can further improve the performance by replacing level-1 BLAS like operations with matrix multiplications. The main part of the full-blocked one-sided Jacobi method (OSBJ) is computing the SVD of a pair of block columns of the input matrix. Thus, the computation method of this partial SVD is important for both accuracy and performance of OSBJ. Hari proposed three methods for this computation, and we found out that one of the method called “V2”, which computes the QR decomposition in this partial SVD using the Cholesky QR method, is the fastest and has comparable accuracy with other method. This is interesting considering that Cholesky QR is generally known as fast but unstable algorithm. In this article, we analyze the accuracy of V2 and explain why and when the Cholesky QR method used in it can compute the QR decomposition accurately. We also show the performance and accuracy comparisons with other computational methods.