CGLS-GCV: A hybrid algorithm for low-rank-deficient problems. An algorithm is given for solving two rank-deficient problems: construction of a minimum norm solution of the unperturbed least squares problem for a linear system and computation of approximations of the column (row) space of the coefficient matrix. The algorithm’s attribute is the satisfactory detection of the rank even if the singular spectrum has no clear gap. It relies on a combination of the conjugate-gradient method for least squares with regularization in the generated Krylov subspace. The algorithm is designed to avoid the singular value decomposition (SVD). Its accuracy is comparable to SVD but at a lower computational cost. An example from magnetic resonance spectroscopy is used as an illustration.

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