A New Proof on the Growth Factor in Gaussian Elimination for Generalized Higham Matrices

Authors: Qian-Ping Guo, Hou-Biao Li

Abstract:

The generalized Higham matrix is a complex symmetric matrix A = B + iC, where both B ∈ Cn×n and C ∈ Cn×n are Hermitian positive definite, and i = √−1 is the imaginary unit. The growth factor in Gaussian elimination is less than 3√2 for this kind of matrices. In this paper, we give a new brief proof on this result by different techniques, which can be understood very easily, and obtain some new findings.

Keywords: CSPD matrix, positive definite, Schur complement, Higham matrix, Gaussian elimination, Growth factor.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1091600

References:

[1] A. George, Kh. D. Ikramov, A. B. Kucherov, On the growth factor in Gaussian elimination for generalized Higham matrices, Numer. Lin. Alg. Appl., 9 (2002) 107–114 .
[2] N. J. Higham, Factorizing complex symmetric matrices with positive real and imaginary parts, Math. Comput., 67 (1998) 1591–1599.
[3] N. J. Higham, Accuracy and stablility of numerical algorithms (2nd), Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2002.
[4] U. V. Rienen. Numerical Methods in Computational Electrodynamics: Linear Systems in Practical Applications. Number 12 in Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin, 2001.
[5] A. George, K. D. Ikramov, On the growth factor in Gaussian elimination for matrices with sharp angular field of values, Calcolo, 41 (2004) 27–36.
[6] Kh. D. Ikramov, A. B. Kucherov, Bounding the growth factor in Gaussian elimination for Buckley’s class of complex symmetric matrices, Numer. Lin. Alg. Appl., 7 (2000) 269–274.
[7] Kh. D. Ikramov, Determinantal inequalities for accretive-dissipative matrices, J. Math. Sci., 121 (2004) 2458–2464.
[8] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.