Incomplete factorization methods for systems with sparse matrices (Q1323973)

A class of invertible square matrices is defined, where the restricted matrices fulfil an exponential estimation. This class is called exponentially closed, if, with the matrix \(A\), also the inverse of \(A\) fulfils certain estimations. On the basis of such an exponentially closed class of matrices a theory of estimates of the elements of inverse matrices and the elements of factors of orthogonal and triangular factorizations is constructed for a wide class of sparse matrices, e.g. generating band matrices, diagonally sparse generating matrices, lattice sparse matrices, and band sparse generating matrices. The results are applied to the incomplete factorization method as the methods of artificially restricting filling-ins, the methods of positional elimination, and the barrier method. Estimations are given for the difference of the factors of the matrix factorization and the incomplete factorization. This paper gives a mathematical justification for solving big sparse linear systems by incomplete factorization methods.