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Low-rank matrix approximation using the Lanczos bidiagonalization process with applications. (English) Zbl 0962.65038
Approximations $B$ (of rank $j$) of a given large non-square matrix $A$ are studied. Their optimal form is defined as a minimizer of the Frobenius norm of the difference $A-B$. In principle, it can be constructed via a $j$-dimensional part (projection) of a certain diagonalized form of $A$. In such a setting one would need the full singular value decomposition of $A$ (which generalizes the current diagonalization using two orthogonal matrices $P$ and $Q$). 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.

MSC:
 65F20 Overdetermined systems, pseudoinverses (numerical linear algebra) 65F50 Sparse matrices (numerical linear algebra)
Software:
Regularization tools; svdpack