Bounds for the Rayleigh quotient and the spectrum of self-adjoint operators
| Autor: | Zhu, Peizhen, Argentati, Merico E., Knyazev, Andrew V. |
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| Rok vydání: | 2012 |
| Předmět: | |
| Zdroj: | SIAM Journal on Matrix Analysis and Applications 2013 34:1, 244-256 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1137/120884468 |
| Popis: | The absolute change in the Rayleigh quotient (RQ) is bounded in this paper in terms of the norm of the residual and the change in the vector. If $x$ is an eigenvector of a self-adjoint bounded operator $A$ in a Hilbert space, then the RQ of the vector $x$, denoted by $\rho(x)$, is an exact eigenvalue of $A$. In this case, the absolute change of the RQ $|\rho(x)-\rho(y)|$ becomes the absolute error in an eigenvalue $\rho(x)$ of $A$ approximated by the RQ $\rho(y)$ on a given vector $y.$ There are three traditional kinds of bounds of the eigenvalue error: a priori bounds via the angle between vectors $x$ and $y$; a posteriori bounds via the norm of the residual $Ay-\rho(y)y$ of vector $y$; mixed type bounds using both the angle and the norm of the residual. We propose a unifying approach to prove known bounds of the spectrum, analyze their sharpness, and derive new sharper bounds. The proof approach is based on novel RQ vector perturbation identities. Comment: 13 pages |
| Databáze: | arXiv |
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