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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.

65F20Overdetermined systems, pseudoinverses (numerical linear algebra)
65F50Sparse matrices (numerical linear algebra)