2015
DOI: 10.4208/cicp.260514.231214a
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Computing the Smallest Eigenvalue of Large Ill-Conditioned Hankel Matrices

Abstract: This paper presents a parallel algorithm for finding the smallest eigenvalue of a particular form of ill-conditioned Hankel matrix, which requires the use of extremely high precision arithmetic. Surprisingly, we find that commonly-used approaches that are designed for high efficiency are actually less efficient than a direct approach under these conditions. We then develop a parallel implementation of the algorithm that takes into account the unusually high cost of individual arithmetic operations. Our approac… Show more

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Cited by 12 publications
(27 citation statements)
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“…(3.14) and the numerically computed also decrease very slowly with N . This was indeed observed in [9]. The differences were decreasing with N , however, much slower than for β = 1/2, and, as a result, the numerics did not convincingly confirm eq.…”
Section: Saddle Point Approximationmentioning
confidence: 78%
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“…(3.14) and the numerically computed also decrease very slowly with N . This was indeed observed in [9]. The differences were decreasing with N , however, much slower than for β = 1/2, and, as a result, the numerics did not convincingly confirm eq.…”
Section: Saddle Point Approximationmentioning
confidence: 78%
“…We would like to stress that the algorithm we use in this paper was chosen to be as numerically stable as possible (see comparison between Secant, Householder, Jacobi, Lanczos in [9] and the Section 5.2 on page 19), and the numerical challenges are intrinsic to the problem and are not due to instabilities of the algorithms.…”
Section: Condition Numbermentioning
confidence: 99%
“…Let λ N denote the smallest eigenvalue of HN. The asymptotic behavior of λ N for large N has been investigated . Also see Beckermann and Lubinsky,in which the authors have studied the behavior of the condition number κ()HN:=ΛNλN, where Λ N denotes the largest eigenvalue of HN.…”
Section: Introductionmentioning
confidence: 99%
“…And, finally, in Section , we present a comparison of the theoretical results to numeric calculations for the smallest eigenvalue, for various values of α , β , and N . The numerical computations were performed using the parallel algorithms developed in Emmart et al…”
Section: Introductionmentioning
confidence: 99%
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