A Padé family of iterations for the matrix sector function and the matrix pth root

Abstract

A family of iterations for the sector function based on the Padé table of a certain hypergeometric function is derived and investigated. This generalizes a result of Kenney and Laub for the sign function and yields a whole family of iterative methods for computing the matrix pth root.It is proved that the iterations for the matrix sector function corresponding to the main diagonal of the Padé table preserve the structure of a group of automorphisms associated with a scalar product.The regions of convergence of the Padé iterations for the matrix sector function are investigated theoretically and experimentally. Certain conjectures formulated on the regions of convergence have been verified in particular cases.

Iteration functions for pth roots of complex numbers

“…This has already been done for the particular case of Newton's method ( j = 2), as we recall in the next theorem (see also [16,Theorem 4.5] for a similar result).…”

“…Moreover, replacing w −1 by w in (2.11), we can observe that the iterations L j coincide with the iteration functions addressed firstly in [21], and later on in [15]. L j is also a particular case of the Padé family of iterations studied in [16] (see more details in Section 4).…”

A binary powering Schur algorithm for computing primary matrix roots

“…According to (14), for k = 1, we have that 16 ii , and that V (2) jj = U 4 jj . Moreover, from Lemma 1 it follows that…”

“…In order to have a general algorithm, some kind of preprocessing of the matrix A should be done. The first general and stable algorithm was given by Iannazzo [12] and some others have followed [3,4,13,16]. The computational cost of these algorithms is O(n 3 log 2 p) arithmetic operations (ops) and the storage required is O(n 2 log 2 p) real numbers.…”

A Schur–Padé Algorithm for Fractional Powers of a Matrix

“…Various methods are available for the qth root problem, based on the Schur decomposition and appropriate recurrences [14], [37], Newton or inverse Newton iterations [15], [27], Padé iterations [28], [33], or a variety of other techniques [6]; see [23,Chap. 7] and [25] for surveys.…”