An analysis on the efficiency of Euler’s method for computing the matrix \(p\)th root. (English) Zbl 1424.65062

Ling, Yonghui; Huang, Zhengda

Numer. Linear Algebra Appl. 24, No. 6, e2104, 21 p. (2017).

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 C.-H. Guo in 2010 [Linear Algebra Appl. 432, No. 8, 1905–1922 (2010; Zbl 1190.65065)]. 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.

MSC:

65F60

Numerical computation of matrix exponential and similar matrix functions

Keywords:

convergence; Euler’s method; matrix \(p\)th root; Schur decomposition

Software:

Matlab