A Study of Algorithms for the p th Root of Matrix


Authors : Langote Ulhas Baban; Dr. Mulay Prashant P.

Volume/Issue : Volume 9 - 2024, Issue 9 - September


Google Scholar : https://tinyurl.com/5n7ncdv9

Scribd : https://tinyurl.com/ybsbu7u4

DOI : https://doi.org/10.38124/ijisrt/IJISRT24SEP1314

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Abstract : Some results for Pth root of square matrix are revived. It shows that matrix sign function and Wiener- Hopf factorization plays important role in Pth root of matrix. Some new algorithms for computing P th root numerically can design by these results. We can analyze Stability properties of iterative methods for convergence.

Keywords : P th Root Of Matrix, Matrix Sign Function, Newton’s Method, Cyclic Reduction, Wiener – Hopf Factorization, Graeffe- Iteration, Laurent Polynomial.

References :

  1. P. Benner, R. Byers, V. Mehrmann and H. Xu, A unified deflating subspace approach for classes of polynomial and rational matrix equations, Preprint SFB393/00-05, Zentrum für Technomathematik, Universität Bremen, Bremen, Germany (January 2000).
  2. M.A. Hasan, J.A.K. Hasan and L. Scharenroich, New integral representations and algorithms for computing nth roots and the matrix sector function of nonsingular complex matrices, in: Proc. of the 39th IEEE Conf. on Decision and Control, Sydney, Australia (2000) pp. 4247–4252.
  3. N.J. Higham, Newton’s method for the matrix square root, Math. Comp. 46(174) (1986) 537–549.
  4. 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.       
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  6. M.I. Smith, A Schur algorithm for computing matrix pth roots, SIAM J. Matrix Anal. Appl. 24(4) (2003) 971–989.
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  8. Y.T. Tsay, L.S. Shieh and J.S.H. Tsai, A fast method for computing the principal nth roots of complex matrices, Linear Algebra Appl. 76 (1986) 205–221.

Some results for Pth root of square matrix are revived. It shows that matrix sign function and Wiener- Hopf factorization plays important role in Pth root of matrix. Some new algorithms for computing P th root numerically can design by these results. We can analyze Stability properties of iterative methods for convergence.

Keywords : P th Root Of Matrix, Matrix Sign Function, Newton’s Method, Cyclic Reduction, Wiener – Hopf Factorization, Graeffe- Iteration, Laurent Polynomial.

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