A VARIANT OF SIMPLIFIED NEWTON’S METHOD WITH NUMERICAL STABILITY TO CALCULATE THE SQUARE ROOT OF A MATRIX
Autor: Gonzalo Moroyoqui Estrella
Coautor(es): Alfredo Mendoza Mexía (1) Adriana Leticia Navarro Verdugo (1)
Gonzalo Moroyoqui Estrella (2)
1 Universidad de Sonora Unidad Regional Sur, Navojoa Sonora México
2 Universidad de Sonora Unidad Centro, Hermosillo Sonora México
To solve the matrix equation F(X) ≡ X^2-A = 0 involves calculating the square root X of A. Newton's method solves the previous problem but it is not attractive due to its excessive computational cost. The method of Newton is simplified and a pair of iterations, that are convergent, are obtained. Nevertheless, Nicholas Highman demonstrated its numerical instability by means of the analysis of disturbances and numerical examples. This fact restricts its application in practical problems. In this article Newton’s simplified method is modified, and its numerical instability is eliminated to calculate the square root of a matrix. The modification consists in factoring A in each iteration as the product of two matrices Bk y Ck that commute between them (Bk Ck = Ck Bk) and with Xk (Bk Xk Ck = Ck Xk Bk), thereby, the method becomes attractive because it is robust, convergent, computationally economical and, for practical purposes, it is numerically stable.
Keywords: Newton's simplified method, square root of a matrix, successive factorization, convergent, numerically stable.