Conditioning analysis of positive definite matrices by approximate factorizations. (English) Zbl 0678.65029

The conditioning analysis of positive definite matrices by approximate LU factorizations is usually reduced to that of Stieltjes matrices by means of perturbation arguments like spectral equivalence. The authors define “almost Stieltjes” matrices \(A:=A_ 0+A_ 2,\) where \(A_ 0\) is a Stieltjes matrix while \(A_ 2\) is both nonnegative and nonnegative definite. They show that this class of matrices can be used as a reference class for the conditioning analysis. This is advantageous especially for the conditioning analysis of finite element approximations of large multidimensional steady-state diffusion problems.
Reviewer: N.K√∂ckler


65F35 Numerical computation of matrix norms, conditioning, scaling
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations


Full Text: DOI


[1] Axelsson, O., A generalized SSOR method, Bit, 13, 443-467, (1972) · Zbl 0256.65046
[2] Axelsson, O.; Barker, V.A., Finite element solution of boundary value problems: theory and computation, (1984), Academic Press New York · Zbl 0537.65072
[3] Beauwens, R., Upper eigenvalue bounds for pencils of matrices, Linear algebra appl., 62, 87-104, (1984) · Zbl 0575.65029
[4] Beauwens, R., On Axelsson’s perturbations, Linear algebra appl., 68, 221-242, (1985) · Zbl 0599.65023
[5] Beauwens, R., Lower eigenvalue bounds for pencils of matrices, Linear algebra appl., 85, 101-119, (1987) · Zbl 0613.15011
[6] George, A.; Liu, J.W., Computer solution of large sparse positive definite systems, (1981), Prentice-Hall Englewood Cliffs, NJ · Zbl 0516.65010
[7] Gustafsson, I., A class of first order factorization methods, Bit, 18, 142-156, (1978) · Zbl 0386.65006
[8] Gustafsson, I., Stability and rate of convergence of modified incomplete Cholesky factorization methods, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.