# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Error bounds on approximate solutions to systems of linear algebraic equationsHTML articles powered by AMS MathViewer

by A. de la Garza
Math. Comp. 7 (1953), 81-84 Request permission
References
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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