##
**Computing the square roots of a class of circulant matrices.**
*(English)*
Zbl 1264.65064

Summary: We first investigate the structures of the square roots of a class of circulant matrices and give classifications of the square roots of these circulant matrices. Then, we develop several algorithms for computing their square roots. We show that our algorithms are faster than the standard algorithm which is based on the Schur decomposition.

### MSC:

65F30 | Other matrix algorithms (MSC2010) |

### Software:

mftoolbox
PDF
BibTeX
XML
Cite

\textit{Y. Mei}, J. Appl. Math. 2012, Article ID 647623, 15 p. (2012; Zbl 1264.65064)

Full Text:
DOI

### References:

[1] | P. J. Davis, Circulant Matrices, Chelsea Publishing, New York, NY, USA, 2nd edition, 1994. · Zbl 0898.15021 |

[2] | R. M. Gray, Toeplitz and Circulant Matrices: A Review, Stanford University Press, Stanford, Calif, USA, 2000. |

[3] | A. Mayer, A. Castiaux, and J.-P. Vigneron, “Electronic Green scattering with n-fold symmetry axis from block circulant matrices,” Computer Physics Communications, vol. 109, no. 1, pp. 81-89, 1998. · Zbl 0938.81560 |

[4] | R. E. Cline, R. J. Plemmons, and G. Worm, “Generalized inverses of certain Toeplitz matrices,” Linear Algebra and its Applications, vol. 8, pp. 25-33, 1974. · Zbl 0273.15004 |

[5] | G. R. Argiroffo and S. M. Bianchi, “On the set covering polyhedron of circulant matrices,” Discrete Optimization, vol. 6, no. 2, pp. 162-173, 2009. · Zbl 1203.90127 |

[6] | N. L. Tsitsas, E. G. Alivizatos, and G. H. Kalogeropoulos, “A recursive algorithm for the inversion of matrices with circulant blocks,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 877-894, 2007. · Zbl 1125.65026 |

[7] | S. Shen and J. Cen, “On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 2891-2897, 2010. · Zbl 1211.15029 |

[8] | Z. L. Jiang and Z. B. Xu, “A new algorithm for computing the inverse and generalized inverse of the scaled factor circulant matrix,” Journal of Computational Mathematics, vol. 26, no. 1, pp. 112-122, 2008. · Zbl 1174.65016 |

[9] | S. G. Zhang, Z. L. Jiang, and S. Y. Liu, “An application of the Gröbner basis in computation for the minimal polynomials and inverses of block circulant matrices,” Linear Algebra and its Applications, vol. 347, pp. 101-114, 2002. · Zbl 1005.65040 |

[10] | N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2008. · Zbl 1167.15001 |

[11] | G. W. Cross and P. Lancaster, “Square roots of complex matrices,” Linear and Multilinear Algebra, vol. 1, pp. 289-293, 1974. · Zbl 0283.15008 |

[12] | C. R. Johnson, K. Okubo, and R. Reams, “Uniqueness of matrix square roots and an application,” Linear Algebra and its Applications, vol. 323, no. 1-3, pp. 51-60, 2001. · Zbl 0976.15009 |

[13] | M. A. Hasan, “A power method for computing square roots of complex matrices,” Journal of Mathematical Analysis and Applications, vol. 213, no. 2, pp. 393-405, 1997. · Zbl 0893.65028 |

[14] | C. B. Lu and C. Q. Gu, “The computation of the square roots of circulant matrices,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6819-6829, 2011. · Zbl 1229.65068 |

[15] | K. D. Ikramov, “Hamiltonian square roots of skew-Hamiltonian matrices revisited,” Linear Algebra and its Applications, vol. 325, no. 1-3, pp. 101-107, 2001. · Zbl 0978.15010 |

[16] | R. Reams, “Hadamard inverses, square roots and products of almost semidefinite matrices,” Linear Algebra and its Applications, vol. 288, no. 1-3, pp. 35-43, 1999. · Zbl 0933.15006 |

[17] | G. H. Golub and C. F. van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, Md, USA, 3rd edition, 1996. · Zbl 0865.65009 |

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.