Year: 2002
Author: Gui-Zhi Chen, Zhong-Xiao Jia
Journal of Computational Mathematics, Vol. 20 (2002), Iss. 5 : pp. 519–524
Abstract
Let A be a real square matrix and $V^TAV = G$ be an upper Hessenberg matrix with positive subdiagonal entries, where $V$ is an orthogonal matrix. Then the implicit $Q$-theorem states that once the first column of $V$ is given then $V$ and $G$ are uniquely determined. In this paper, three results are established. First, it holds a reverse order implicit $Q$-theorem: once the last column of $V$ is given, then $V$ and $G$ are uniquely determined too. Second, it is proved that for a Krylov subspace two formulations of the Arnoldi process are equivalent and in one to one correspondence. Finally, by the equivalence relation and the reverse order implicit $Q$-theorem, it is proved that for the Krylov subspace, if the last vector of vector sequence generated by the Arnoldi process is given, then the vector sequence and resulting Hessenberg matrix are uniquely determined.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2002-JCM-8936
Journal of Computational Mathematics, Vol. 20 (2002), Iss. 5 : pp. 519–524
Published online: 2002-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 6
Keywords: Implicit Q-theorem Reverse order implicit Q-theorem Truncated version Arnoldi process.