Numerical stability of the Chebyshev method for the solution of large linear systems |
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Authors: | H Woźniakowski |
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Institution: | (1) Department of Computer Science, Carnegie-Mellon University, Schenley Park, 15213 Pittsburgh, PA, USA;(2) Present address: University of Warsaw, Warsaw, Poland |
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Abstract: | Summary This paper contains the rounding error analysis for the Chebyshev method for the solution of large linear systemsAx+g=0 whereA=A
* is positive definite. We prove that the Chebyshev method in floating point arithmetic is numerically stable, which means that the computed sequence {x
k} approximates the solution such that
x
k
–![agr](/content/trv3225315033832/xxlarge945.gif) is of order ![zeta](/content/trv3225315033832/xxlarge950.gif) A![Verbar](/content/trv3225315033832/xxlarge8214.gif) A
–1![Verbar](/content/trv3225315033832/xxlarge8214.gif) ![Verbar](/content/trv3225315033832/xxlarge8214.gif) ![agr](/content/trv3225315033832/xxlarge945.gif) where is the relative computer precision.We also point out that in general the Chebyshev method is not well-behaved, which means that the computed residualsr
k=Ax
k+g are of order ![zeta](/content/trv3225315033832/xxlarge950.gif) A 2 A
–1![Verbar](/content/trv3225315033832/xxlarge8214.gif) ![Verbar](/content/trv3225315033832/xxlarge8214.gif) ![agr](/content/trv3225315033832/xxlarge945.gif) .This work was supported in part by the Office of Naval Research under Contract N0014-67-0314-0010, NR 044-422 and by the National Science Foundation under Grant GJ32111 |
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Keywords: | |
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