Stable versions of Newton’s iteration for computing the principal matrix pth root $A^{1/p}$ of an n × n matrix A are provided. In the case in which $X_0$ is the identity matrix, it is proved that the method converges for any matrix A having eigenvalues with modulus less than 1 and with positive real parts. Based on these results we provide a general algorithm for computing the principal pth root of any matrix A having no nonpositive real eigenvalues. The algorithm has quadratic convergence, is stable in a neighborhood of the solution, and has a cost of $O(n^3 log p)$ operations per step. Numerical experiments and comparisons are performed.
On the Newton method for the matrix Pth root
IANNAZZO, Bruno
2006
Abstract
Stable versions of Newton’s iteration for computing the principal matrix pth root $A^{1/p}$ of an n × n matrix A are provided. In the case in which $X_0$ is the identity matrix, it is proved that the method converges for any matrix A having eigenvalues with modulus less than 1 and with positive real parts. Based on these results we provide a general algorithm for computing the principal pth root of any matrix A having no nonpositive real eigenvalues. The algorithm has quadratic convergence, is stable in a neighborhood of the solution, and has a cost of $O(n^3 log p)$ operations per step. Numerical experiments and comparisons are performed.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
