Let A be a nonsingular n × n real matrix and let p ≥ 2 be an integer. Suppose &fgr; p ( A , X ) is a function from the n × n real matrices X to themselves and let X 0 be a given matrix. For k ≥ 0 define X k + 1 = &fgr; p ( A , X k ) by iteration. This operation is a hyperpower method if I - AX k + 1 = ( I - AX k ) p for all k > 0. If the spectral radius of I - AX 0 is less than one, the { X k } converges to A - 1 (and conversely), with order of convergence p. The purpose of this paper is to derive error bounds for this hyperpower method (Section 2), then apply these bounds to obtain the inclusion of A - 1 in an interval matrix (Section 3), to establish the monotonicity of the width of these interval matrices (Section 4), and to exhibit a numerical example (Section 5).
The error bound calculation is done for an arbitrary multiplicative matrix norm. This norm is then restricted to be a row or column sum for applications. The author “intervalizes” the function &fgr; p ( A , X ), adding an interval matrix all of whose entries are [ d , d ] where d is derived from A and X, and then iteratively defines a sequence of interval matrices, and widths d k, containing A - 1, from any pair of hyperpower methods with parameters p , r. The order of convergence of { d k } is shown to be pk . The efficiency index of the method is shown to be e ( r , p ) = ( rp ) 1&slash; ( r + 2p ). The author further shows that the width sequence declines monotonically.
This paper is intended for researchers in numerical linear algebra methods and assumes a graduate-level familiarity with the discipline. The style of presentation is that of theoretical mathematics, with formal proofs.
