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

Abstract

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.

On semilocal convergence analysis for two-step Newton method under generalized Lipschitz conditions in Banach spaces

“…with L(t) = c(c − 1)h 2 αt c−2 , where α f (u 0 ) −1 . Therefore, Theorem 1 with c ∈ [1, 2) is applicable, but not Corollary 1, to concluding that the sequence {u k } generated by the two-step Newton method with initial guess 0 = e converges to a local solution of (48).…”

“…Consider the following two-dimensional nonlinear convection-diffusion equation [8,31,70] −(u xx + u yy ) + q 1 u x + q 2 u y = u c for (x, y) ∈ u(x, y) = 0 for (x, y) ∈ (48) where = (0, 1) × (0, 1), with its boundary, q 1 and q 2 are positive constants used to measure the magnitudes of the convective terms, and c ∈ [1,2]. Applying the five-point finite-difference scheme to the diffusive terms and the central difference scheme to the convective terms, respectively, a system of nonlinear equations is…”

“…Kantorovich-like theorem and Smale-like α-theory for these two iterative methods are referred to [10, 11, 21-23, 33, 49] and references therein. For the applications in the field of matrix functions, the efficiency (when properly implemented) of these two iterative methods have been shown in [36,48] for computing matrix pth root and in [54,55] for computing the polar decomposition of a matrix.…”

A study of Schröder’s method for the matrix pth root using power series expansions

“…For a given integer p ≥ 2 and a matrix A ∈ C n×n whose eigenvalues are in the open disk {z ∈ C : |z − 1| < 1}, the principal pth root of A exists and is denoted by A 1/p [10]. Various methods can be used to compute A 1/p ; see [2,7,8,9,10,11,12,13,14,15,18,20,23,24].…”

Semilocal Convergence Analysis for Two-Step Newton Method under Generalized Lipschitz Conditions in Banach Spaces

“…The most important two are the Euler method and the Halley method; see for example [9, 10, 20-22, 32, 48] and references therein. For the applications in the field of matrix functions, the efficiency (when properly implemented) of these two methods have been shown in [35,47] for computing matrix pth root and in [52,53] for computing the polar decomposition of a matrix. Another more general family of the cubic extensions is the family of Euler-Halley methods type methods in Banach spaces, which includes the Euler and the Halley method as its special cases and has been studied extensively in [29,31].…”