Further Study of Some Iterative Methods for the Matrix Pth Root

Abstract

In this thesis, we solve some problems arising in the study of some iterative methods for finding the principal pth root of a matrix. Our main interest is in Newton's method, Halley's method and Chebyshev's method. We also study Schroder's method (which includes Newton's method and Chebyshev's method as special cases). The study of these methods for the matrix case can be reduced to the study in the scalar case. Some theoretical properties of the iterative methods can be obtained by examining the Taylor series expansion of the iterates. It has been observed by Guo that the Taylor coefficients have a favorable sign pattern for Newton's method and Halley's method, and that some nice theoretical results can be proved after the conjectured sign pattern is confirmed. In this thesis, we are going to prove this conjecture raised by Guo. The key idea of the proof is to establish a monotonicity property for the coefficients as the iteration progresses. Using this idea, we prove the sign pattern of Newton's method and Halley's method without too much difficulty. We then move on to prove a similar sign pattern for Schroder's method. The proof is much more complicated, but the basic strategy remains the same. After we have confirmed the sign pattern of Taylor coefficients for Newton's method, Halley's method and Schroder's method, we present some related results for the matrix case. In this thesis, we also establish a convergence region for Newton's method that is much larger than previously available ones.