Behavior of the Steepest Descent Method in Minimizing Rayleigh Quotient
Summary :
In this paper we discuss the limiting behavior of the search direction of the steepest descent method in minimizing the Rayleigh quotient. This minimization problem is equivalent to finding the smallest eigenvalue of a matrix. It is shown that the search direction asymptotically alternates between two directions represented by linear combinations of two eigenvectors of the matrix. This is similar to the phenomenon in minimizing the quadratic form. We also show that these eigenvectors correspond to the largest and second-smallest eigenvalues, unlike in the case of the quadratic form.
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- IEICE TRANSACTIONS on Fundamentals Vol.E80-A No.1 pp.176-182
- Publication Date
- 1997/01/25
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- Category
- Numerical Analysis and Optimization
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Takashi OZEKI, Taizo IIJIMA, "Behavior of the Steepest Descent Method in Minimizing Rayleigh Quotient" in IEICE TRANSACTIONS on Fundamentals,
vol. E80-A, no. 1, pp. 176-182, January 1997, doi: .
Abstract: In this paper we discuss the limiting behavior of the search direction of the steepest descent method in minimizing the Rayleigh quotient. This minimization problem is equivalent to finding the smallest eigenvalue of a matrix. It is shown that the search direction asymptotically alternates between two directions represented by linear combinations of two eigenvectors of the matrix. This is similar to the phenomenon in minimizing the quadratic form. We also show that these eigenvectors correspond to the largest and second-smallest eigenvalues, unlike in the case of the quadratic form.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e80-a_1_176/_p
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@ARTICLE{e80-a_1_176,
author={Takashi OZEKI, Taizo IIJIMA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Behavior of the Steepest Descent Method in Minimizing Rayleigh Quotient},
year={1997},
volume={E80-A},
number={1},
pages={176-182},
abstract={In this paper we discuss the limiting behavior of the search direction of the steepest descent method in minimizing the Rayleigh quotient. This minimization problem is equivalent to finding the smallest eigenvalue of a matrix. It is shown that the search direction asymptotically alternates between two directions represented by linear combinations of two eigenvectors of the matrix. This is similar to the phenomenon in minimizing the quadratic form. We also show that these eigenvectors correspond to the largest and second-smallest eigenvalues, unlike in the case of the quadratic form.},
keywords={},
doi={},
ISSN={},
month={January},}
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TY - JOUR
TI - Behavior of the Steepest Descent Method in Minimizing Rayleigh Quotient
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 176
EP - 182
AU - Takashi OZEKI
AU - Taizo IIJIMA
PY - 1997
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E80-A
IS - 1
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - January 1997
AB - In this paper we discuss the limiting behavior of the search direction of the steepest descent method in minimizing the Rayleigh quotient. This minimization problem is equivalent to finding the smallest eigenvalue of a matrix. It is shown that the search direction asymptotically alternates between two directions represented by linear combinations of two eigenvectors of the matrix. This is similar to the phenomenon in minimizing the quadratic form. We also show that these eigenvectors correspond to the largest and second-smallest eigenvalues, unlike in the case of the quadratic form.
ER -
English
