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Mathematics of Computation

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Error bounds on approximate solutions to systems of linear algebraic equations


Author: A. de la Garza
Journal: Math. Comp. 7 (1953), 81-84
MSC: Primary 65.0X
DOI: https://doi.org/10.1090/S0025-5718-1953-0054340-8
MathSciNet review: 0054340
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  • [1] The referee makes the following comments: ``If all $ \vert{d_{ij}}\vert > 0$, the optimum choice of $ \gamma $ is unique and is the eigenvector $ {u_1}$ of $ \alpha (D)$ whose components are all positive, and $ k$ is then the dominant eigenvalue $ {\lambda _1}$ of $ \alpha (D)$. This follows from a lemma that, since all $ {g_i} > 0,{\lambda _1}$ lies strictly between the minimum and maximum of the ratios $ {e'_i}\alpha (D)\gamma /{e'_i}\gamma $, unless the ratios are all equal (and hence equal to $ {\lambda _1}$). The lemma is a slight extension of Theorem I of Hazel Perfect, 'On matrices with positive elements,' Quart. Jn. of Math., s. 2, v. 2, 1951, p. 286-290. ``The vector $ \alpha ^{p}(D)e$, which is asymptotically a multiple of $ {u_1}$ as $ p \to \infty$, may be a useful approximation to $ \gamma $ for sufficiently large $ p$."

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DOI: https://doi.org/10.1090/S0025-5718-1953-0054340-8
Article copyright: © Copyright 1953 American Mathematical Society