Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Preconditioning for Accurate Solutions of Linear Systems and Eigenvalue
  Problems   This paper develops the preconditioning technique as a method to address the
accuracy issue caused by ill-conditioning. Given a preconditioner $M$ for an
ill-conditioned linear system $Ax=b$, we show that, if the inverse of the
preconditioner $M^{-1}$ can be applied to vectors accurately, then the linear
system can be solved accurately. A stability concept called inverse-equivalent
accuracy is introduced to describe higher accuracy that is achieved and an
error analysis will be presented. As an application, we use the preconditioning
approach to accurately compute a few smallest eigenvalues of certain
ill-conditioned matrices. Numerical examples are presented to illustrate the
error analysis and the performance of the methods.

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The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
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The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
1
The categories of this paper are: cs
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-ex