التفاصيل البيبلوغرافية
| العنوان: |
BOUNDS FOR THE RAYLEIGH QUOTIENT AND THE SPECTRUM OF SELF-ADJOINT OPERATORS. |
| المؤلفون: |
PEIZHEN ZHU, ARGENTATI, MERICO E., KNYAZEV, ANDREW V. |
| المصدر: |
SIAM Journal on Matrix Analysis & Applications; 2013, Vol. 34 Issue 1, p244-256, 13p |
| مصطلحات موضوعية: |
MATHEMATICAL bounds, RAYLEIGH quotient, SPECTRUM analysis, SELFADJOINT operators, HILBERT space, ERROR analysis in mathematics |
| مستخلص: |
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 ρ(x), is an exact eigenvalue of A. In this case, the absolute change of the RQ |ρ(x) - ρ(y)| becomes the absolute error for an eigenvalue ρ(x) of A approximated by the RQ ρ(y) on a given vector y. There are three traditional kinds of bounds for eigenvalue errors: a priori bounds via the angle between vectors x and y; a posteriori bounds via the norm of the residual Ay -ρ(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. [ABSTRACT FROM AUTHOR] |
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