Abstract
Some iterative methods are introduced and demonstrated for finding the matrix sign function. It is analytically shown that the new schemes are asymptotically stable. Convergence analysis along with the error bounds of the main proposed method is established. Different numerical experiments are employed to compare the behavior of the new schemes with the existing matrix iterations of the same type.
Highlights
The theory of matrix functions becomes an active topic of research in the field of advanced numerical linear algebra
This paper is concerned with a special case known as matrix sign function, which is of clear importance in the theory of matrix functions [7]
In the theory of matrix functions, many of the matrix functions could effectively be calculated by the existing iterative methods for finding the solution of nonlinear equations [14]
Summary
The theory of matrix functions becomes an active topic of research in the field of advanced numerical linear algebra (see, e.g., [1,2,3,4]). Let us, as Higham considered in the fifth Chapter of [6], assume throughout this paper that the matrix A ∈ Cn×n has no eigenvalues on the imaginary axis This assumption implies that the matrix sign function,. In 1991, a fundamental family of matrix iterations for finding the matrix sign function S was introduced in [10] using Padeapproximants to f(ξ) = (1 − ξ)−1/2 and the following characterization: sign (z). The well-known Halley’s matrix iteration of order three can be deduced as follows: Xk+1 = [I + 3Xk2] [Xk (3I + Xk2)]−1. To find a method with fourth order and global convergence, we give another matrix iteration therein.
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