An algorithm to improve nearly orthonormal sets of vectors on a vector processor

Abstract. The symmetric orthogonalization, which is obtained from the polar decomposition of a matrix, is optimal. We propose an iterative algorithm to compute this orthogonalization on vector computers. It is especially efficient when the original matrix is near an orthonormal matrix. Key words, polar decomposition, iterative method, square root, vector computer AMS(MOS) subject classification. 65F25 Introduction. In the computation of the eigenvectors of a Hermitian matrix, it is necessary to check the orthonormality ofthe computed vectors, since for close eigenvalues there is an accompanying loss of orthogonality. Usually, especially when the vectors have been computed by inverse iteration, the Gram-Schmidt orthonormalization is performed on the groups ofeigenvectors corresponding to close eigenvalues. If the residual is checke