{ "id": "1807.04251", "version": "v1", "published": "2018-07-11T17:14:46.000Z", "updated": "2018-07-11T17:14:46.000Z", "title": "A study of SchrÃ¶der's method for the matrix \$p\$th root using power series expansions", "authors": [ "Chun-Hua Guo", "Di Lu" ], "categories": [ "math.NA" ], "abstract": "When \$A\$ is a matrix with all eigenvalues in the disk \$|z-1|<1\$, the principal \$p\$th root of \$A\$ can be computed by Schr\\\"oder's method, among many other methods. In this paper we present a further study of Schr\\\"oder's method for the matrix \$p\$th root, through an examination of power series expansions of some sequences of scalar functions. Specifically, we obtain a new and informative error estimate for the matrix sequence generated by the Schr\\\"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.", "revisions": [ { "version": "v1", "updated": "2018-07-11T17:14:46.000Z" } ], "analyses": { "keywords": [ "power series expansions", "th root", "schrÃ¶ders method", "monotonic convergence result", "real nonsingular" ], "note": { "typesetting": "TeX", "pages": 0, "language": "en", "license": "arXiv", "status": "editable" } } }