• Kim, Hyun-Min ;
  • Kim, Young-jin ;
  • Meng, Jie
  • Received : 2017.12.23
  • Accepted : 2018.06.01
  • Published : 2018.11.01


The matrix equation $X^p+A^*XA=Q$ has been studied to find the positive definite solution in several researches. In this paper, we consider fixed-point iteration and Newton's method for finding the matrix p-th root. From these two considerations, we will use the Newton-Schulz algorithm (N.S.A). We will show the residual relation and the local convergence of the fixed-point iteration. The local convergence guarantees the convergence of N.S.A. We also show numerical experiments and easily check that the N.S. algorithm reduce the CPU-time significantly.


fixed-point iteration;Newton's method;Newton-Schulz algorithm;local convergence


  1. B. Iannazzo, On the Newton method for the matrix pth root, SIAM J. Matrix Anal. Appl. 28 (2006), no. 2, 503-523.
  2. B. Iannazzo, A family of rational iterations and its application to the computation of the matrix pth root, SIAM J. Matrix Anal. Appl. 30 (2008/09), no. 4, 1445-1462.
  3. Z. Jia and M. Wei, Solvability and sensitivity analysis of polynomial matrix equation $X^s\;+\;A^TX^tA\;=\;Q$, Appl. Math. Comput. 209 (2009), no. 2, 230-237.
  4. C. Kenney and A. J. Laub, Condition estimates for matrix functions, SIAM J. Matrix Anal. Appl. 10 (1989), no. 2, 191-209.
  5. H.-M. Kim, Y.-J. Kim, and J.-H. Seo, Local convergence of functional iterations for solving a quadratic matrix equation, Bull. Korean Math. Soc. 54 (2017), no. 1, 199-214.
  6. J. Meng and H.-M. Kim, The positive definite solution to a nonlinear matrix equation, Linear Multilinear Algebra 64 (2016), no. 4, 653-666.
  7. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, reprint of the 1970 original, Classics in Applied Mathematics, 30, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
  8. A. C. M. Ran and M. C. B. Reurings, On the nonlinear matrix equation $X+A^{\ast}{\mathfrak{F}}(X)A\;=\;Q$: solutions and perturbation theory, Linear Algebra Appl. 346 (2002), 15-26.
  9. A. C. M. Ran and M. C. B. Reurings, The symmetric linear matrix equation, Electron. J. Linear Algebra 9 (2002), 93-107.
  10. A. C. M. Ran and M. C. B. Reurings, A nonlinear matrix equation connected to interpolation theory, Linear Algebra Appl. 379 (2004), 289-302.
  11. M. C. B. Reurings, Contractive maps on normed linear spaces and their applications to nonlinear matrix equations, Linear Algebra Appl. 418 (2006), no. 1, 292-311.
  12. M. C. B. Reurings, Symmetric matrix equations, Ph.D. thesis, Vrije Universiteit Amsterdam, 2003;
  13. M. I. Smith, A Schur algorithm for computing matrix pth roots, SIAM J. Matrix Anal. Appl. 24 (2003), no. 4, 971-989.
  14. R. A. Smith, Infinite product expansions for matrix n-th roots, J. Austral. Math. Soc. 8 (1968), 242-249.
  15. D. A. Bini, N. J. Higham, and B. Meini, Algorithms for the matrix pth root, Numer. Algorithms 39 (2005), no. 4, 349-378.
  16. S. H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, Approximating the logarithm of a matrix to specified accuracy, SIAM J. Matrix Anal. Appl. 22 (2001), no. 4, 1112-1125.
  17. S. M. El-Sayed and A. C. M. Ran, On an iteration method for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl. 23 (2001/02), no. 3, 632-645.
  18. J. C. Engwerda, A. C. M. Ran, and A. L. Rijkeboer, Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation $X+A^{\ast}X^{-1}A\;=\;Q$, Linear Algebra Appl. 186 (1993), 255-275.
  19. C.-H. Guo, On Newton's method and Halley's method for the principal pth root of a matrix, Linear Algebra Appl. 432 (2010), no. 8, 1905-1922.
  20. C.-H. Guo and N. J. Higham, A Schur-Newton method for the matrix pth root and its inverse, SIAM J. Matrix Anal. Appl. 29 (2007), 396-412.
  21. W. D. Hoskins and D. J. Walton, A faster, more stable method for computing the pth roots of positive definite matrices, Linear Algebra Appl. 26 (1979), 139-163.