Rounding error analysis of block orthogonalization in the one-sided block Jacobi SVD method

Abstract We provide the error analysis for one method of orthogonalization of matrix column blocks in floating point arithmetic, which is a crucial step in the one-sided block Jacobi algorithm for computing the singular value decomposition of a general matrix. The orthogonalization is based on computing the Gram matrix and its Cholesky decomposition to obtain the R-factor. Then, the one-sided element-wise Jacobi algorithm is applied to compute the singular value decomposition of the R-factor via Givens rotations, and, finally, the column block is updated by accumulated Givens rotations. We provide the upper bounds for roundoff errors in each of above mentioned steps, and our analysis is based on a row and column scaling of matrices arising during computation both in exact and floating point arithmetic. Our main result is the upper bound for the orthogonality error of computed left singular vectors of a given matrix column block, the form of which is discussed in detail. Numerical experiments illustrate the developed theory.