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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
MathSciNet review: 0054340
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    The referee makes the following comments: “If all $|{d_{ij}}| > 0$, 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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Article copyright: © Copyright 1953 American Mathematical Society