Computing the Matrix Geometric Mean of Two HPD Matrices: A Stable Iterative Method
محورهای موضوعی : مجله بین المللی ریاضیات صنعتی
F. Kiyoumarsi
1
(Department of Mathematics, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran.)
کلید واژه: Stability, Sign function, HPD, Convergence, Iterative methods,
چکیده مقاله :
A new iteration scheme for computing the sign of a matrix which has no pure imaginary eigenvalues is presented. Then, by applying a well-known identity in matrix functions theory, an algorithm for computing the geometric mean of two Hermitian positive definite matrices is constructed. Moreover, another efficient algorithm for this purpose is derived free from the computation of principal matrix square root. Finally, several experiments are collected.
در این مقاله، یک روش تکراری کارا برای محاسبه علامت یک ماتریس که هیچ مقدار ویژه ی موهومی ندارد ارائه می شود. سپس با استفاده از یک همسانی شناخته شده در تئوری عملکرد ماتریس ها، یک الگوریتم برای محاسبه میانگین هندسی از دو ماتریس هرمیتی معین مثبت بدست می آید. علاوه بر این یک الگوریتم کارامد دیگر برای این هدف ارائه میشود که وابسته به جذر ماتریس نباشد. وسرانجام چند آزمایش هم برای نمایش کاربرد آن انجام خواهد شد.
[1] S. Amat, J. A. Ezquerro, M. A. Hern´andezVer´on, On a new family of high-order iterative methods for the matrix pth root, Numer. Linear Algebra Appl. 22 (2015) 585-595.
[2] R. Bhatia, Positive definite matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, (2007).
[3] E. Denman, A. Beavers, The matrix sign function and computations in systems, Appl. Math. Comput. 2 (1976), 63-94.
[4] A. Frommer, B. Hashemi, T. Sablik, Computing enclosures for the inverse square root and the sign function of a matrix, Linear Algebra Appl. 456 (2014) 199-213.
[5] O. Gomilko, D. B. Karp, M. Lin, K. Zietak, Regions of convergence of a Pad´e family of iterations for the matrix sector function and the matrix p-th root, J. Comput. Appl. Math. 236 (2012) 4410-4420.
[6] F. Greco, B. Iannazzo, F. Poloni, The Pad´e iterations for the matrix sign function and their reciprocals are optimal, Linear Algebra Appl. 436 (2012) 472-477.
[7] N. J. Higham, The matrix sign decomposition and its relation to the polar decomposition, Lin. Algebra Appl. 212 (1994) 3-20.
[8] N. J. Higham, Accuracy and Stability of Numerical Algorithms, Second Edition, SIAM, UK, 2002.
[9] N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008.
[10] B. Iannazzo, The geometric mean of two matrices from a computational viewpoint,Numer. Linear Algebra Appl. 23 (2016) 208-229.
[11] C. S. Kenney, A. J. Laub, Rational iterative methods for the matrix sign function, SIAM J. Matrix Anal. Appl. 12 (1991) 273-291.
[12] F. Khaksar Haghani, F. Soleymani, An improved Schulz-type iterative method for matrix inversion with application, Tran. Inst. Measur. Cont. 36 (2014) 983-991.
[13] D. Li, P. Liu, J. Kou, An improvement of Chebyshev-Halley methods free from second derivative, Appl. Math. Comput. 235 (2014) 221-225.
[14] T. Lotfi, F. Soleymani, M. Ghorbanzadeh, P. Assari, On the construction of some triparametric iterative methods with memory, Numer. Algorithms 70 (2015) 835-845.
[15] J. D. Lawson, Y. Lim, The geometric mean, matrices, metrics and more, Amer. Math. Month. 108 (2001) 797-812.
[16] M. Sh. Misrikhanov, V. N. Ryabchenko, Matrix sign function in the problems of analysis and design of the linear systems, Auto. Remote Cont. 69 (2008) 198-222.
[17] G. Pusz, S. L. Woronowicz, Functional calculus for sesquilinear forms and the purification map, Rep. Math. Phys. 8 (1975) 159-170.
[18] F. Soleymani, E. Tohidi, S. Shateyi, F. Khaksar Haghani, Some matrix iterations for computing matrix sign function, J. Appl. Math. (2014) 9 pages.
[19] F. Soleymani, P. S. Stanimirovi´c, I. Stojanovi´c, A novel iterative method for polar decomposition and matrix sign function, Disc. Dyn. Nat. Soc. (2015).
[20] F. Soleymani, M. Sharifi, S. Shateyi, F. Khaksar Haghani, An algorithm for computing geometric mean of two Hermitian positive definite matrices via matrix sign, Abst. Appl. Anal. Volume 2014, Article ID 978629,
[21] J. L. Varona, Graphic and numerical comparison between iterative methods, Math. Intell. 24 (2002) 37-46.
[22] Wolfram Research, Inc., Mathematica, Version 10.1, Champaign, IL (2015).
