Un método simplificado de newton para calcular la raíz cuadrada de una matriz real simétrica definida positiva

Latest revision as of 11:02, 14 June 2017

Abstract

One of the numerically stable methods to calculate he Squire root of a matrix is the Newton’s Method, but it has the disadvantage that in each iteration it is necessary to solve a Sylvester’s matrix equation for full matrix, which is computationally expensive. Modifications to the method have been made, resulting a pair of simplified iterations, but it has been proved, through the perturbations theory, that these are numerically unstable. In this article, a simplified alternative iteration of Newton’s Method for the calculation of the square root of a positive definite real matrix is presented, which is attractive for being convergent, computationally economic and, for practical purposes, numerically stable.

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 == Abstract ==
-Uno de los métodos numéricos estables más utilizado para calcular la raíz cuadrada de una matriz es el Método de Newton (MN) el cual tiene la desventaja de que, em cada iteración, es necesario resolver una ecuación matricial de Sylvester para matrices densas resultando computacionalmente costos. Se han hecho simplificaciones al MN, resultando un par de iteraciones simplificadas. Higham demostró, a través de la teoría de perturbaciones, que éstas son numéricamente inestables. En este artículo se desarrolla una iteración alternativa sinplificada del Método de Newton (IASMN) para calcular la raíz cuadrada de una matriz real simétrica definida positiva, la cual es atractiva por ser convergente, computacionalmente económica y, para propósitos prácticos, es numéricamente estable. Summary One of the numerically stable methods to calculate he Squire root of a matrix is the Newton’s Method, but it has the disadvantage that in each iteration it is necessary to solve a Sylvester’s matrix equation for full matrix, which is computationally expensive. Modifications to the method have been made, resulting a pair of simplified iterations, but it has been proved, through the perturbations theory, that these are numerically unstable. In this article, a simplified alternative iteration of Newton’s Method for the calculation of the square root of a positive definite real matrix is presented, which is attractive for being convergent, computationally economic and, for practical purposes, numerically stable.
+One of the numerically stable methods to calculate he Squire root of a matrix is the Newton’s Method, but it has the disadvantage that in each iteration it is necessary to solve a Sylvester’s matrix equation for full matrix, which is computationally expensive. Modifications to the method have been made, resulting a pair of simplified iterations, but it has been proved, through the perturbations theory, that these are numerically unstable. In this article, a simplified alternative iteration of Newton’s Method for the calculation of the square root of a positive definite real matrix is presented, which is attractive for being convergent, computationally economic and, for practical purposes, numerically stable.
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