Computing the square roots of matrices with central symmetry. (English) Zbl 1121.65045

Liu, Zhongyun; Zhang, Yulin; Rui, Ralha

Appl. Math. Comput. 186, No. 1, 715-726 (2007).

The reduced forms of centrosymmetric, skew-centrosymmetric and centro-Hermitian matrices are exploited to study the structure of square roots of such matrices and to design algorithms for computing those square roots. A new structured algorithm to compute the square root is proposed that is approximately 5.5 times cheaper than the standard one. In the case of centro-Hermitian matrices, the corresponding structured algorithm is approximately eight times cheaper than the standard one. The stability and the accuracy of the algorithms proposed are discussed.

Reviewer: Petko Hr. Petkov (Sofia)

MSC:

65F30	Other matrix algorithms (MSC2010)
15A24	Matrix equations and identities

Keywords:

matrix square root; central symmetry; Schur algorithm; centrosymmetric; skew-centrosymmetric; centro-Hermitian matrices; algorithms; stability

References:

[1]	Andrew, A. L., Eigenvectors of certain matrices, Linear Alg. Appl., 7, 151-162 (1973) · Zbl 0255.65021
[2]	Bartels, R. H.; Stewart, G. W., Solution of equation \(AX + XB =C\), Comm. ACM, 15, 820-826 (1972) · Zbl 1372.65121
[3]	Björck, A.; Hammarling, S., A Schur method for the square root of a matrix, Linear Alg. Appl., 52/53, 127-140 (1983) · Zbl 0515.65037
[4]	Cadzow, J. A., Signal enhancement – a composite property mapping algorithm, IEEE Trans. Acoust. Speech Signal Proc., 36, 49-62 (1988) · Zbl 0649.93059
[5]	Cross, G. W.; Lancaster, P., Square roots of complex matrices, Linear Multi-linear Alg., 1, 289-293 (1974) · Zbl 0283.15008
[6]	Datta, L.; Morgera, S., On the reducibility of centrosymmetric matrices applications in engineering problems, Circ. Syst. Signal Process, 8, 71-96 (1989) · Zbl 0674.15005
[7]	Davis, P. I.; Higham, N. J., A Parlett-Schur algorithm for computing matrix functions, SIAM J. Matrix Anal. Appl., 25, 464-485 (2003) · Zbl 1052.65031
[8]	Faßbender, H.; Ikramov, K. D., Computing matrix-vector products with centrosymmetric and centrohermitian matrices, Linear Alg. Appl., 364, 235-241 (2003) · Zbl 1016.65022
[9]	Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), Johns Hopkins University Press: Johns Hopkins University Press Baltimore · Zbl 0865.65009
[10]	Higham, N. J., Computing real square roots of a real matrix, Linear Alg. Appl., 88/89, 405-430 (1987) · Zbl 0625.65032
[11]	Higham, N. J.; Mackey, D. S.; Mackey, N.; Tisseur, F., Function preserving matrix groups and iterations for the matrix square root, SIAM J. Matrix Anal. Appl., 26, 849-877 (2005) · Zbl 1079.65053
[12]	Hill, R. D.; Betas, R. G.; Waters, S. R., On centrohermitian matrices, SIAM J. Matrix Anal. Appl., 11, 128-133 (1990) · Zbl 0709.15021
[13]	Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0729.15001
[14]	Lee, A., Centrohermitian and skew-centrohermitian matrices, Linear Alg. Appl., 29, 205-210 (1980) · Zbl 0435.15019
[15]	Li, H.; Stoica, P.; Li, J., Computationally efficient maximum likelihood estimation of structured covariance matrices, IEEE Trans. Signal Proc., 47, 1314-1323 (1999) · Zbl 0986.62014
[16]	Liu, Z. Y., Some properties of centrosymmetric matrices, Appl. Math. Comput., 141, 17-26 (2002)
[17]	Liu, Z.; Chen, H.; Cao, H., The computation of the principal square roots of centrosymmetric \(h\)-matrices, Appl. Math. Comp., 175, 319-329 (2006) · Zbl 1093.65044
[18]	Meyer, C. D., Matrix Analysis and Applied Linear Algebra (2000), SIAM: SIAM Philadelphia
[20]	Parlett, B. N., A recurrence among the elements of functions of triangular matrices, Linear Alg. Appl., 14, 117-121 (1976) · Zbl 0353.65027
[21]	Smith, M. I., A Schur algorithm for computing matrix \(p\) th roots, SIAM J. Matrix Anal. Appl., 24, 971-989 (2003) · Zbl 1040.65038
[22]	Weaver, J., Centrosymmetric (cross-symmetric) matrices their basic properties eigenvalues and eigenvectors, Amer. Math. Monthly, 92, 711-717 (1985) · Zbl 0619.15021