The original paper is in English. Non-English content has been machine-translated and may contain typographical errors or mistranslations. ex. Some numerals are expressed as "XNUMX".
Copyrights notice
The original paper is in English. Non-English content has been machine-translated and may contain typographical errors or mistranslations. Copyrights notice
고유값 문제에 대한 Jacobi-Davidson 방법과 Riccati 방법을 연구합니다. 방법에서는 반복당 수정 방정식이라는 비선형 방정식을 풀어야 하며, 방정식을 어떻게 푸느냐에 따라 두 방법의 차이가 발생합니다. Jacobi-Davidson/Riccati 방법에서 수정 방정식은 선형화 여부에 관계없이 해결됩니다. 문헌에서는 선형화를 피하는 것이 방정식의 더 나은 해를 얻고 더 빠른 수렴을 가져오기 위한 개선으로 알려져 있습니다. 실제로 Riccati 방법은 일부 문제에 대해 우수한 수렴 거동을 보여주었습니다. 그럼에도 불구하고 Riccati 방법의 장점은 여전히 불분명합니다. 왜냐하면 수정 방정식이 정확하지는 않지만 정확도가 낮기 때문입니다. 본 논문에서는 수정 방정식의 근사해를 분석하고 Riccati 방법이 고유값 문제의 특정 해를 계산하는 데 특화되어 있다는 점을 명확히 했습니다. 결과는 타겟 솔루션에 따라 두 가지 방법을 선택적으로 사용해야 함을 시사한다. 우리의 분석은 수치 실험을 통해 검증되었습니다.
Takafumi MIYATA
Fukuoka Institute of Technology
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부
Takafumi MIYATA, "On Correction-Based Iterative Methods for Eigenvalue Problems" in IEICE TRANSACTIONS on Fundamentals,
vol. E101-A, no. 10, pp. 1668-1675, October 2018, doi: 10.1587/transfun.E101.A.1668.
Abstract: The Jacobi-Davidson method and the Riccati method for eigenvalue problems are studied. In the methods, one has to solve a nonlinear equation called the correction equation per iteration, and the difference between the methods comes from how to solve the equation. In the Jacobi-Davidson/Riccati method the correction equation is solved with/without linearization. In the literature, avoiding the linearization is known as an improvement to get a better solution of the equation and bring the faster convergence. In fact, the Riccati method showed superior convergence behavior for some problems. Nevertheless the advantage of the Riccati method is still unclear, because the correction equation is solved not exactly but with low accuracy. In this paper, we analyzed the approximate solution of the correction equation and clarified the point that the Riccati method is specialized for computing particular solutions of eigenvalue problems. The result suggests that the two methods should be selectively used depending on target solutions. Our analysis was verified by numerical experiments.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E101.A.1668/_p
부
@ARTICLE{e101-a_10_1668,
author={Takafumi MIYATA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={On Correction-Based Iterative Methods for Eigenvalue Problems},
year={2018},
volume={E101-A},
number={10},
pages={1668-1675},
abstract={The Jacobi-Davidson method and the Riccati method for eigenvalue problems are studied. In the methods, one has to solve a nonlinear equation called the correction equation per iteration, and the difference between the methods comes from how to solve the equation. In the Jacobi-Davidson/Riccati method the correction equation is solved with/without linearization. In the literature, avoiding the linearization is known as an improvement to get a better solution of the equation and bring the faster convergence. In fact, the Riccati method showed superior convergence behavior for some problems. Nevertheless the advantage of the Riccati method is still unclear, because the correction equation is solved not exactly but with low accuracy. In this paper, we analyzed the approximate solution of the correction equation and clarified the point that the Riccati method is specialized for computing particular solutions of eigenvalue problems. The result suggests that the two methods should be selectively used depending on target solutions. Our analysis was verified by numerical experiments.},
keywords={},
doi={10.1587/transfun.E101.A.1668},
ISSN={1745-1337},
month={October},}
부
TY - JOUR
TI - On Correction-Based Iterative Methods for Eigenvalue Problems
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1668
EP - 1675
AU - Takafumi MIYATA
PY - 2018
DO - 10.1587/transfun.E101.A.1668
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E101-A
IS - 10
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - October 2018
AB - The Jacobi-Davidson method and the Riccati method for eigenvalue problems are studied. In the methods, one has to solve a nonlinear equation called the correction equation per iteration, and the difference between the methods comes from how to solve the equation. In the Jacobi-Davidson/Riccati method the correction equation is solved with/without linearization. In the literature, avoiding the linearization is known as an improvement to get a better solution of the equation and bring the faster convergence. In fact, the Riccati method showed superior convergence behavior for some problems. Nevertheless the advantage of the Riccati method is still unclear, because the correction equation is solved not exactly but with low accuracy. In this paper, we analyzed the approximate solution of the correction equation and clarified the point that the Riccati method is specialized for computing particular solutions of eigenvalue problems. The result suggests that the two methods should be selectively used depending on target solutions. Our analysis was verified by numerical experiments.
ER -