) of a given large non-square matrix
are studied. Their optimal form is defined as a minimizer of the Frobenius norm of the difference
. In principle, it can be constructed via a
-dimensional part (projection) of a certain diagonalized form of
. In such a setting one would need the full singular value decomposition of
(which generalizes the current diagonalization using two orthogonal matrices
). The paper offers a cheaper algorithm based on the Lanczos bidiagonalization. The authors perform the error and stopping analysis and demonstrate that a mere one-sided re-orthogonalization of this process guarantees a sufficient precision. A collection of matrices from several application areas is used in illustrative tests.