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An analysis on the efficiency of Euler's method for computing the matrix p th root
Author(s) -
Ling Yonghui,
Huang Zhengda
Publication year - 2017
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2104
Subject(s) - mathematics , invertible matrix , identity matrix , matrix (chemical analysis) , euler's formula , convergence (economics) , preconditioner , eigenvalues and eigenvectors , root (linguistics) , schur decomposition , band matrix , backward euler method , symmetric matrix , schur complement , square matrix , mathematical analysis , iterative method , pure mathematics , algorithm , euler equations , linguistics , materials science , quantum mechanics , economics , composite material , economic growth , physics , philosophy
Summary It is shown that the matrix sequence generated by Euler's method starting from the identity matrix converges to the principal p th 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.