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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Asymptotic behavior of vector recurrences with applications

Authors: Alan Feldstein and J. F. Traub
Journal: Math. Comp. 31 (1977), 180-192
MSC: Primary 65Q05
MathSciNet review: 0426464
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Abstract: The behavior of the vector recurrence $ {{\mathbf{y}}_{n + 1}} = M{{\mathbf{y}}_n} + {{\mathbf{w}}_{n + 1}}$ is studied under very weak assumptions. Let $ \lambda (M)$ denote the spectral radius of M and let $ \lambda (M) \geqslant 1$. Then if the $ {{\mathbf{w}}_n}$ are bounded in norm and a certain subspace hypothesis holds, the root order of the $ {{\mathbf{y}}_n}$ is shown to be $ \lambda (M)$. If one additional hypothesis on the dimension of the principal Jordan blocks of M holds, then the quotient order of the $ {{\mathbf{y}}_n}$ is also $ \lambda (M)$. The behavior of the homogeneous recurrence is studied for all values of $ \lambda (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.

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Article copyright: © Copyright 1977 American Mathematical Society