Böttcher, Albrecht ; Potts, Daniel : Probability against condition number and sampling of multivariate trigonometric
- Author(s):
-
Böttcher, Albrecht
Potts, Daniel
- Title:
- Probability against condition number and sampling of multivariate trigonometric
- Electronic source:
-
application/pdf
- Preprint series:
- Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 2, 2006
- Mathematics Subject Classification:
-
65F35 [ Matrix norms, conditioning, scaling ] 15A12 [ Conditioning of matrices ] 47B35 [ Toeplitz operators, Hankel operators, Wiener-Hopf operators ] 60H25 [ Random operators and equations ] 94A20 [ Sampling theory ] - Abstract:
- The difficult term in the condition number $\| A\n \, \| A^{-1}\|$ of a large linear system $Ap=y$ is the spectral norm of $A^{-1}$. To eliminate this term, 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 $\| \de p\|$ and $\| \de y\|$ 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.
- Keywords:
- condition number, probability argument, linear system, Toeplitz matrix, nonuniform sampling,
multivariate trigonometric polynomial
- Language:
-
English
- Publication time:
- 3 / 2006
