Abstract
It is shown how the mid-point iterative method with cubical rate of convergence can be applied for finding the principal matrix square root. Using an identity between matrix sign function and matrix square root, we construct a variant of mid-point method which is asymptotically stable in the neighborhood of the solution. Finally, application of the presented approach is illustrated in solving a matrix differential equation.
Highlights
Introductory NotesLet us consider a scalar function f and a matrix A ∈ Cn×n and specify f(A) to be a matrix function of the same dimensions as A
We first apply the following mid-point cubically convergent scheme of Frontini and Sormani
In order to derive a higher order new version of midpoint method for computing square root, we use some of the methods in the literature and obtain a new scheme for A1/2 as well
Summary
Let us consider a scalar function f and a matrix A ∈ Cn×n and specify f(A) to be a matrix function of the same dimensions as A. We are interested in numerical computation of matrix square root, which is one of the most fundamental matrix functions with potential applications. Denman and Beavers (DB) in [6] proposed a quadratically convergent variant of Newton’s matrix method (3) which is numerically stable in what follows: Y0 = A, Z0 = I, k = 0, 1, . We first apply the following mid-point cubically convergent scheme of Frontini and Sormani (given for scalar nonlinear equations [9]).
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