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Diagonally compensated reduction and related preconditioning methods
Author(s) -
Axelsson O.,
Kolotilina L.
Publication year - 1994
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1680010207
Subject(s) - mathematics , diagonally dominant matrix , finite element method , coefficient matrix , matrix (chemical analysis) , discretization , reduction (mathematics) , factorization , convergence (economics) , positive definite matrix , sparse matrix , matrix splitting , block matrix , diagonal , symmetric matrix , mathematical analysis , algorithm , state transition matrix , pure mathematics , geometry , eigenvalues and eigenvectors , physics , materials science , composite material , quantum mechanics , gaussian , economics , invertible matrix , thermodynamics , economic growth
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.