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When $A$ is a matrix with all eigenvalues in the disk $|z-1|<1$, theprincipal $p$th root of $A$ can be computed by Schr\"oder's method, among manyother methods. In this paper we present a further study of Schr\"oder's methodfor the matrix $p$th root, through an examination of power series expansions ofsome sequences of scalar functions. Specifically, we obtain a new andinformative error estimate for the matrix sequence generated by theSchr\"oder's method, a monotonic convergence result when $A$ is a nonsingular$M$-matrix, and a structure preserving result when $A$ is a nonsingular$M$-matrix or a real nonsingular $H$-matrix with positive diagonal entries.

A study of Schr\"oder's method for the matrix $p$th root using power series expansions