The behavior of the vector recurrence y" + i = Afyn + wn+1 is stud- ied under very weak assumptions. Let \(Af) denote the spectral radius of M and let \(M) > 1. Then if the wn are bounded in norm and a certain subspace hy- pothesis holds, the root order of the yn is shown to be \(M). If one additional hypothesis on the dimension of the principal Jordan blocks of M holds, then the quotient order of the yn is also \(M). The behavior of the homogeneous re- currence is studied for all values of \(M). These results are applied to the analysis of (1) Nonlinear iteration with application to iteration with memory and to parallel iteration algorithms. (2) Order and efficiency of composite iteration. 1. Introduction. We study the behavior of the vector recurrence (i) y"+i =My» +w«+i under very weak assumptions. We apply our results to the analysis of iterations for nonlinear equations and to the composition of such iterations. In particular our re- sults can be used to study one-point iterations with memory and iterations for solving nonlinear equations on parallel computers. An extended discussion of applications, including use of the power method to calculate the spectral radius, may be found in Feldstein and Traub (74).
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