Lower estimates on the condition number of a Toeplitz sinc matrix and related questions

Abstract

As one new result, for a symmetric Toeplitz $ \operatorname{sinc} $ $n \times n$-matrix $A(t)$ depending on a parameter $t$, lower estimates (tending to infinity as t vanishes) on the pertinent condition number are derived. A further important finding is that prior to improving the obtained lower estimates it seems to be more important to determine the lower bound on the parameter $t$ such that the smallest eigenvalue $\mu_n(t)$ of $A(t)$ can be reliably computed since this is a precondition for determining a reliable value for the condition number of the Toeplitz $ \operatorname{sinc} $ matrix. The style of the paper is expository in order to address a large readership.

Keywords

• Condition number
• eigenvalues and eigenvectors
• inverse power method
• power method
• Toeplitz sinc matrix

References

1. D. Hertz: Simple Bounds on the Extreme Eigenvalues of Toeplitz Matrices, IEEE Transactions on Information Theory, 38 (1) (1992), 175–176.
2. N. J. Higham: Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia (1996).
3. G. H. Goloub, Ch. F. van Loan: Matrix Computations, The Johns Hopkins University Press, Baltimore and London (1989).
4. F. Stummel: Diskrete Approximation linear Operatoren. II (Discrete Approximation of Linear Operators. II). Math. Z., 120 (1971), 231–264.
5. F. Stummel, K. Hainer: Introduction to Numerical Analysis (English Translation by E.R. Dawson of the First Edition of the German Original of 1971), Scottish Academic Press, Edinburgh (1980).
6. F. Stummel, K. Hainer: Praktische Mathematik (Introduction to Numerical Analysis), Second Edition, B.G. Teubner, Stuttgart (1982).
7. F. Stummel, L. Kohaupt: Eigenwertaufgaben in Hilbertschen Räumne. Mit Aufgaben und vollständigen Lösungen, (Eigenvalue Problems in Hilbert spaces. With Exercises and Complete Solutions), Logos Verlag, Berlin (2021).
8. J. H. Wilkinson: The Algebraic Eigenvalue Problem, Oxford University Press, Oxford (1965).

9. Y.Wu: On the positiveness of a functional symmetric matrix used in digital filter design, Journal of Circuits, Systems, and Computers, 13 (5) (2004), 1105–1110.
10. Y. Wu, D. H. Mugler: A robust DSP integrator for acceleration signals, IEEE Transactions on Biomedical Engineering, Vol. 51 (2) (2004), 385–389.
11. Y. Wu, N. Sepehri: Interpolation of bandlimited signals from uniform or non-uniform integral samples, Electronic Letters, 47 (1) (2011), 6th Jan.