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Abstract: We propose to precondition the CGILS method based on the incomplete hyperbolic Gram–Schmidt methods for the solution of the large and sparse indefinite least squares (ILS) problem, which requires minimization of an indefinite quadratic form. Numerical experiments show that the incomplete hyperbolic classical/modified Gram–Schmidt (IHCGS/IHMGS) preconditioners can generally greatly reduce the number of iterations, and IHMGS has a better quality for ill-conditioned problems but may be expensive in computation cost. [Copyright &y& Elsevier]

Abstract: The hyperbolic modified Gram–Schmidt (HMGS) method is proposed for block downdating the Cholesky factorization. The method might be unsatisfactory due to rounding errors. A modified version based on the MGS process is presented and is shown to be mixed stable. Numerical tests show that the new method has the same numerical properties as the generalized LINPACK-type algorithm, and can work better than the Householder-based algorithm given by Bojanczyk and Steinhardt (1991) . [ABSTRACT FROM AUTHOR]

A matrix-scaled total least squares problem by scaling some columns of error matrix with a real positive parameter γ is presented. For particularly large or small value of γ , a hybrid algorithm with deflation is proposed to avoid potential instability of standard singular value decomposition. [ABSTRACT FROM AUTHOR]