Block LU factorization is stable for block matrices whose inverses are block diagonally dominant. (English. Russian original) Zbl 1116.65034

Dzhordzh, A.; Ikramov, Kh. D.

J. Math. Sci., New York 127, No. 3, 1962-1968 (2005); translation from Zap. Nauchn. Semin. POMI 296, 15-26 (2003).

Summary: Let \(A\in M_n(C)\) and let the inverse matrix \(B= A^{-1}\) be block diagonally dominant by rows (columns) with respect to an \(m\times m\) block partitioning and a matrix norm. We show that \(A\) possesses a block LU factorization w.r.t. the same block partitioning, and the growth factor for \(A\) in this factorization is bounded above by \(1+\sigma\), where \(\sigma= \max_{1\leq i\leq m}\sigma_i\) and \(\sigma_i\), \(0\leq \sigma_i\leq 1\), are the row (column) block dominance factors of \(B\). Further, the off-diagonal blocks of \(A\) (and of its block Schur complements) satisfy the inequalities \(\|A_{ji} A_{ii}^{-1}\|\leq \sigma_j\), \(j\neq i\).

MSC:

65F05	Direct numerical methods for linear systems and matrix inversion
15A23	Factorization of matrices
65F35	Numerical computation of matrix norms, conditioning, scaling

Software:

mctoolbox

References:

[1]	A. George and Kh. D. Ikramov, ”Gaussian elimination is stable for the inverse of a diagonally dominant matrix,” Math. Comp. (to appear). · Zbl 1050.65024
[2]	N. J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia (1996). · Zbl 0847.65010
[3]	J. W. Demmel, N. J. Higham, and R. S. Schreiber, ”Stability of block LU factorization,” Numer. Linear Algebra Appl., 2, 173–190 (1995). · Zbl 0834.65010 · doi:10.1002/nla.1680020208
[4]	Kh. D. Ikramov, ”A block analogue of the property of diagonal dominance,” Vestn. Mosk. Univ., Ser. XV Vychisl. Mat. Kibern., No. 4, 52–55 (1983). · Zbl 0535.65010