Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



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
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • [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$."

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65.0X

Retrieve articles in all journals with MSC: 65.0X

Additional Information

Article copyright: © Copyright 1953 American Mathematical Society

American Mathematical Society