New algorithms are presented about the principal square root of an matrix . In particular, all the classical iterative algorithms require matrix inversion at every iteration. The proposed inversion free iterative algorithms are based on the Schulz iteration or the Bernoulli substitution as a special case of the continuous time Riccati equation. It is certified that the proposed algorithms are equivalent to the classical Newton method. An inversion free algebraic method, which is based on applying the Bernoulli substitution to a special case of the continuous time Riccati equation, is also proposed. 1. Introduction Let be a real or complex matrix with no eigenvalues on (the closed negative real axis). Then there exists a unique matrix such that and the eigenvalues of lie in the segment . We refer to as the principle square root of . The computation of the principal square root of a matrix is a problem of great interest and practical importance with numerous applications in mathematical and engineering problems. Due to the importance of the problem, many iterative algorithms have been proposed and successfully employed for calculating the principal square root of a matrix [1–7] without seeking the eigenvalues and eigenvectors of the matrix; these algorithms require matrix inversion at every iteration. Blocked Schur Algorithms for Computing the Matrix Square Root are proposed in [8] where the matrix is reduced to upper triangular form and a recurrence relation enables the square root of the triangular matrix to be computed a column or superdiagonal at a time. In this paper new inversion free iterative algorithms for computing the principal square root of a matrix are proposed. The paper is organized as follows: in Section 2 a survey of classical iterative algorithms is presented; all the algorithms use matrix inversion in every iteration. In Section 3, the inversion free iterative algorithms are developed; the algorithms are derived by the Schulz iteration for computing the inversion of a matrix or by eliminating matrix inversion from the convergence criteria of the algorithms. In Section 4, an algebraic method for computing the principal square root of a matrix is presented; the method is associated with the solution of a special case of the continuous time Riccati equation, which arises in linear estimation [9] and requires only one matrix inversion (this is not an iterative method). Simulation results are presented in Section 5. Section 6 summarizes the conclusions. 2. Inversion Use Iterative Algorithms In this section, a survey of classical iterative
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