Semiconvergence of parallel multisplitting methods for symmetric positive semidefinite linear systems. (Q2889388)

The authors present parallel relaxed multisplitting method for solving consistent symmetric positive semidefinite linear systems, based on modified diagonally compensated reduction and incomplete factorizations. The semiconvergence of its three variants is discussed including the generalization of the results for nonsingular linear systems to the singular ones.NEWLINENEWLINEThe parallel multisplitting method is an iterative method based on the construction of matrices' triples \((M,N,E)\) representing the system matrix \(A\) as the difference \(A=M-N\), relaxed two-stage multisplitting then consists of an outer \(A=M-N\) and inner \(M=B-C\) splittings. The matrices \(E\) are diagonal matrices with nonnegative entries, such that their sum is an identity matrix. The matrices \(M\) are invertible and can be constructed, e.g., by means of incomplete Cholesky factorizations corresponding to a symmetric zero pattern set being (semi)convergent for the used relaxation parameter. The convergence is described by the spectral radius of the matrix \(M^{-1}N\), which should be smaller then one. The block diagonal conformable multisplitting of original matrix \(A\) is then such a triple, where the matrices \(M\) and \(E\) have a block diagonal layout. The theory of this method for symmetric positive definite linear systems is extended for symmetric positive semidefinite (i.e., singular) case. The paper contains very nice theoretical results formulated in theorems, lemmas and proofs.