An analysis on the efficiency of Euler's method for computing the matrix pth root

It is shown that the matrix sequence generated by Euler’s method starting from the identity matrix converges to the principal pth root of a square matrix, if all the eigenvalues of the matrix are in a region including the one for Newton’s method given by Guo in 2010. The convergence is cubic if the matrix is invertible. A modification version of Euler’s method using the Schur decomposition is developed. Numerical experiments show that the modified algorithm has the overall good numerical behavior.

[1]  Chun-Hua Guo,et al.  On Newton’s method and Halley’s method for the principal pth root of a matrix , 2010 .

[2]  Krystyna Zietak,et al.  A Padé family of iterations for the matrix sector function and the matrix pth root , 2009, Numer. Linear Algebra Appl..

[3]  Å. Björck,et al.  A Schur method for the square root of a matrix , 1983 .

[4]  Bruno Iannazzo,et al.  On the Newton Method for the Matrix P th Root , 2006, SIAM J. Matrix Anal. Appl..

[5]  Nicholas J. Higham,et al.  A Schur-Padé Algorithm for Fractional Powers of a Matrix , 2011, SIAM J. Matrix Anal. Appl..

[6]  N. Higham.  Computing real square roots of a real matrix , 1987 .

[7]  Nicholas J. Higham,et al.  An Improved Schur-Padé Algorithm for Fractional Powers of a Matrix and Their Fréchet Derivatives , 2013, SIAM J. Matrix Anal. Appl..

[8]  Slobodan Lakić.  On the Computation of the Matrix k-th Root , 1998 .

[9]  Gene H. Golub,et al.  Matrix Computations, Third Edition , 1996 .

[10]  Krystyna Zietak,et al.  The dual Padé families of iterations for the matrix pth root and the matrix p-sector function , 2014, J. Comput. Appl. Math..

[11]  Ana F. Loureiro,et al.  Iteration functions for pth roots of complex numbers , 2011, Numerical Algorithms.

[12]  Minghua Lin.  A residual recurrence for Halley’s method for the matrix pth root , 2010 .

[13]  Bruno Iannazzo,et al.  A Family of Rational Iterations and Its Application to the Computation of the Matrix pth Root , 2008, SIAM J. Matrix Anal. Appl..

[14]  Ana F. Loureiro,et al.  On the convergence of Schröder iteration functions for pth roots of complex numbers , 2011, Appl. Math. Comput..

[15]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[16]  Krystyna Zietak,et al.  Regions of convergence of a Padé family of iterations for the matrix sector function and the matrix pth root , 2012, J. Comput. Appl. Math..

[17]  Bruno Iannazzo,et al.  A binary powering Schur algorithm for computing primary matrix roots , 2010, Numerical Algorithms.

[18]  Nicholas J. Higham,et al.  A Schur-Newton Method for the Matrix \lowercase{\boldmathp}th Root and its Inverse , 2006, SIAM J. Matrix Anal. Appl..

[19]  Awad H. Al-Mohy,et al.  Computing matrix functions , 2010, Acta Numerica.

[20]  Matthew I. Smith,et al.  A Schur Algorithm for Computing Matrix pth Roots , 2002, SIAM J. Matrix Anal. Appl..

[21]  Nicholas J. Higham,et al.  Algorithms for the matrix pth root , 2005, Numerical Algorithms.

[22]  Bruno Iannazzo,et al.  A Schur Logarithmic Algorithm for Fractional Powers of Matrices , 2013, SIAM J. Matrix Anal. Appl..