On Newton's method and Halley's method for the principal $p$th root of a matrix

Abstract

If $A$ is a matrix with no negative real eigenvalues and all zero eigenvalues of $A$ are semisimple, the principal $p$th root of $A$ can be computed by Newton's method or Halley's method, with a preprocessing procedure if necessary. We prove a new convergence result for Newton's method, and discover an interesting property of Newton's method and Halley's method in terms of series expansions. We explain how the convergence of Newton's method and Halley's method can be improved when the eigenvalues of $A$ are known or when $A$ is a singular matrix. We also prove new results on $p$th roots of $M$-matrices and $H$-matrices, and consider the application of Newton's method and Halley's method to find the principal $p$th roots of these special matrices.