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The problem of solving large-scale systems of linear algebraic equations arises in a wide range of applications. In many cases the preconditioned iterative method is a method of choice. This paper deals with the approximate inverse preconditioning AINV/SAINV based on the incomplete generalized Gram–Schmidt process. This type of the approximate inverse preconditioning has been repeatedly used for matrix diagonalization in computation of electronic structures but approximating inverses is of an interest in parallel computations in general. Our approach uses adaptive dropping of the matrix entries with the control based on the computed intermediate quantities. Strategy has been introduced as a way to solve di cult application problems and it is motivated by recent theoretical results on the loss of orthogonality in the generalized Gram– Schmidt process. Nevertheless, there are more aspects of the approach that need to be better understood. The diagonal pivoting based on a rough estimation of condition numbers of leading principal submatrices can sometimes provide inefficient preconditioners. This short study proposes another type of pivoting, namely the pivoting that exploits incremental condition estimation based on monitoring both direct and inverse factors of the approximate factorization. Such pivoting remains rather cheap and it can provide in many cases more reliable preconditioner. Numerical examples from real-world problems, small enough to enable a full analysis, are used to illustrate the potential gains of the new approach.

Abstract: We present a roundoff error analysis of the method for solving the linear least squares problem with full column rank matrix A, using only factors Σ and V from the SVD decomposition of . This method (called here) is an analogue of the method of seminormal equations ( ), where the solution is computed from using only the factor R from the QR factorization of A. Such methods have practical applications when A is large and sparse and if one needs to solve least squares problems with the same matrix A and multiple right-hand sides. However, in general both and are not forward stable. We analyze one step of fixed precision iterative refinement to improve the accuracy of the method. We show that, under the condition , this method (called ) produces a forward stable solution, where denotes the condition number of the matrix A and u is the unit roundoff. However, for problems with only it is generally not forward stable, and has similar numerical properties to the corresponding method. Our forward error bounds for the are slightly better than for the since the terms are not present. We illustrate our analysis by numerical experiments. [Copyright &y& Elsevier]