Böttcher, Albrecht; Potts, Daniel

Probability against condition number and sampling of multivariate trigonometric random polynomials

ETNA, Electron. Trans. Numer. Anal. 26, 178-189, electronic only (2007)

Summary

Summary: The difficult factor in the condition number of a large linear system is the $\textcent $###$\sterling $###$\textcent $###$\textcent $###$\sterling \ddot $§$\copyright \textcent \sterling $ spectral norm of . To eliminate this factor, we here replace worst case analysis by a probabilistic argument. To $\sterling $###§$\copyright $be more precise, we randomly take from a ball with the uniform distribution and show that then, with a certain probability close to one, the relative errors and satisfy with a constant that involves $\textcent $###$\textcent \textcent $###$ !\textcent \textcent $###$\textcent $#"% textcent$###$ !textcentonly the Frobenius and spectral norms of . The success of this argument is demonstrated for Toeplitz systems and £for the problem of sampling multivariate trigonometric polynomials on nonuniform knots. The limitations of the argument are also shown.

Mathematics Subject Classification

65F35, 15A12, 47B35, 60H25, 94A20

Keywords/Phrases

condition number, probability argument, linear system, Toeplitz matrix, nonuniform sampling, multivariate trigonometric polynomial

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