When solving linear algebraic equations with large and sparse coefficient matrices, arising, for instance, from the discretization of partial differential equations, it is quite common to use preconditioning to accelerate the convergence of a basic iterative scheme. Incomplete factorizations and sparse approximate inverses can provide efficient preconditioning methods but their existence and convergence theory is based mostly on M-matrices (H-matrices). In some application areas, however, the arising coefficient matrices are not H-matrices. This is the case, for instance, when higher-order finite element approximations are used, which is typical for structural mechanics problems. We show that modification of a symmetric, positive definite matrix by reduction of positive offdiagonal entries and diagonal compensation of them leads to an M-matrix. This diagonally compensated reduction can take place in the whole matrix or only at the current pivot block in a recursive incomplete factorization method. Applications for constructing preconditioning matrices for finite element matrices are described.
[1]
B. Polman.
Incomplete blockwise factorizations of (block) H-matrices
,
1987
.
[2]
Owe Axelsson,et al.
A robust preconditioner based on algebraic substructuring and two-level grids
,
1989
.
[3]
L. Kolotilina,et al.
On a family of two-level preconditionings of the incomplete block factorization type
,
1986
.
[4]
Owe Axelsson,et al.
A Preconditioned Conjugate Gradient Method for Finite Element Equations, which is Stable for Rounding Errors
,
1980,
IFIP Congress.
[5]
J. Gillis,et al.
Matrix Iterative Analysis
,
1961
.
[6]
I. Gustafsson,et al.
Preconditioning and two-level multigrid methods of arbitrary degree of approximation
,
1983
.
[7]
James M. Ortega,et al.
Iterative solution of nonlinear equations in several variables
,
2014,
Computer science and applied mathematics.
[8]
L. Kolotilina,et al.
Factorized Sparse Approximate Inverse Preconditionings I. Theory
,
1993,
SIAM J. Matrix Anal. Appl..