Improving eigenvectors in Arnoldi's method

Jia ZX, Elsner L (2000)
JOURNAL OF COMPUTATIONAL MATHEMATICS 18(3): 265-276.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
The Ritz vectors obtained by Arnoldi's method may not be good approximations and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the efficiency of Arnoldi type algorithms, we propose a strategy that uses Ritz values obtained from an m-dimensional Krylov subspace but chooses modified approximate eigenvectors in an (m + 1)-dimensional Krylov subspace. Residual norm of each new,approximate eigenpair is minimal over the span of the Ritz vector and the (m + 1)th basis vector, which is available when the m-step Arnoldi process is run. The resulting modified m-step Arnoldi method is better than the standard m-step one in theory and cheaper than the standard (m + 1)-step one. Based on this strategy, we present a modified m-step restarted Arnoldi algorithm. Numerical examples show that the modified m-step restarted algorithm and its version with Chebyshev acceleration are often considerably more efficient than the standard (m + 1)-step restarted ones.
Stichworte
large unsymmetric; the m-step Arnoldi; method; eigenvalue; Ritz value; eigenvector; Ritz vector; modified; the m-step Arnoldi process
Erscheinungsjahr
2000
Zeitschriftentitel
JOURNAL OF COMPUTATIONAL MATHEMATICS
Band
18
Ausgabe
3
Seite(n)
265-276
ISSN
0254-9409
Page URI
https://pub.uni-bielefeld.de/record/1619572

Zitieren

Jia ZX, Elsner L. Improving eigenvectors in Arnoldi's method. JOURNAL OF COMPUTATIONAL MATHEMATICS. 2000;18(3):265-276.
Jia, Z. X., & Elsner, L. (2000). Improving eigenvectors in Arnoldi's method. JOURNAL OF COMPUTATIONAL MATHEMATICS, 18(3), 265-276.
Jia, Z. X., and Elsner, L. (2000). Improving eigenvectors in Arnoldi's method. JOURNAL OF COMPUTATIONAL MATHEMATICS 18, 265-276.
Jia, Z.X., & Elsner, L., 2000. Improving eigenvectors in Arnoldi's method. JOURNAL OF COMPUTATIONAL MATHEMATICS, 18(3), p 265-276.
Z.X. Jia and L. Elsner, “Improving eigenvectors in Arnoldi's method”, JOURNAL OF COMPUTATIONAL MATHEMATICS, vol. 18, 2000, pp. 265-276.
Jia, Z.X., Elsner, L.: Improving eigenvectors in Arnoldi's method. JOURNAL OF COMPUTATIONAL MATHEMATICS. 18, 265-276 (2000).
Jia, ZX, and Elsner, Ludwig. “Improving eigenvectors in Arnoldi's method”. JOURNAL OF COMPUTATIONAL MATHEMATICS 18.3 (2000): 265-276.

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