We extend graph embedding techniques for bounding the spectral condition number of preconditioned systems involving symmetric, irreducibly diagonally dominant M-matrices to systems where the preconditioner is not diagonally dominant. In particular, this allows us to bound the spectral condition number when the preconditioner is based on an incomplete factorization. We provide a review of previo...

This paper presents a preconditioning method based on combining two-sided permutations with a multilevel approach. The nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix. The method can be regarded as a complete pivoting version of the incomplete LU factorization. This leads to an effective incomplete factorizati...

The efficient solution of the normal equations corresponding to a large sparse linear least squares problem can be extremely challenging. Robust incomplete factorization (RIF) preconditioners represent one approach that has the important feature of computing an incomplete LLT factorization of the normal equations matrix without having to form the normal matrix itself. The right-looking implemen...

We describe a novel technique for computing a sparse incomplete factorization of a general symmetric positive de nite matrix A. The factorization is not based on the Cholesky algorithm (or Gaussian elimination), but on A-orthogonalization. Thus, the incomplete factorization always exists and can be computed without any diagonal modi cation. When used in conjunction with the conjugate gradient a...

Limited-memory incomplete Cholesky factorizations can provide robust preconditioners for sparse symmetric positive-definite linear systems. In this paper, the focus is on extending the approach to sparse symmetric indefinite systems in saddle-point form. A limited-memory signed incomplete Cholesky factorization of the form LDL is proposed, where the diagonal matrix D has entries ±1. The main ad...

Incomplete LDL factorizations sometimes produce an indefinite preconditioner evenwhen the input matrix is Hermitian positive definite. The two most popular iterative solvers for symmetric systems, CG and MINRES, cannot use such preconditioners; they require a positive definite preconditioner. One approach, that has been extensively studied to address this problem is to force positive definitene...

The Finite Element modelling of geological faults by penalty contact elements may give rise to illconditioned stiffness matrices with the convergence of iterative solvers accelerated, or even allowed for, by the development and implementation of appropriate preconditioners. The present communication investigates the performance of three different block preconditioners in a realistic geomechanic...

In this article, we present several new permutations for I-matrices making these more suitable for incomplete LDU-factorization preconditioners used in solving linear systems by iterative methods. A general matrix can be transformed by row permutation as well as row and columns scaling into an I-matrix, i.e. a matrix having elements of modulus 1 on the diagonal and elements of modulus of no mor...

We design a grid based multilevel incomplete LU preconditioner (GILUM) for solving general sparse matrices. This preconditioner combines a high accuracy ILU factorization with an algebraic multilevel recursive reduction. The GILUM precondi-tioner is a compliment to the domain based multilevel block ILUT preconditioner. A major diierence between these two preconditioners is the way that the coar...

Convergence acceleration by preconditioning is usually essential when solving the standard least squares problems by an iterative method. IMGS, is an incomplete modiied version of Gram-Schmidt orthogonalization to obtain an incomplete orthogonal factorization pre-conditioner M = R, where A = Q R + E is an approximation of a QR factorization, Q is an orthogonal matrix and R is upper triangular m...