Volume 26, pp. 178-189, 2007.

Probability against condition number and sampling of multivariate trigonometric random polynomials

Albrecht Böttcher and Daniel Potts


The difficult factor in the condition number $\| A\| \, \| A^{-1}\|$ of a large linear system $Ap=y$ is the spectral norm of $A^{-1}$. To eliminate this factor, we here replace worst case analysis by a probabilistic argument. To be more precise, we randomly take $p$ from a ball with the uniform distribution and show that then, with a certain probability close to one, the relative errors $\| \Delta p\|$ and $\| \Delta y\|$ satisfy $\| \Delta p\| \le C \| \Delta y\|$ with a constant $C$ that involves only the Frobenius and spectral norms of $A$. 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.

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Key words

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

AMS subject classifications

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

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