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
São estudados o método Jacobi-Davidson e o método Riccati para problemas de autovalores. Nos métodos, é preciso resolver uma equação não linear chamada equação de correção por iteração, e a diferença entre os métodos vem de como resolver a equação. No método Jacobi-Davidson/Riccati a equação de correção é resolvida com/sem linearização. Na literatura, evitar a linearização é conhecido como uma melhoria para obter uma melhor solução da equação e trazer convergência mais rápida. Na verdade, o método Riccati apresentou comportamento de convergência superior para alguns problemas. No entanto, a vantagem do método Riccati ainda não é clara, porque a equação de correção não é resolvida exatamente, mas com baixa precisão. Neste artigo, analisamos a solução aproximada da equação de correção e esclarecemos que o método Riccati é especializado para calcular soluções particulares de problemas de autovalores. O resultado sugere que os dois métodos devem ser usados seletivamente dependendo das soluções alvo. Nossa análise foi verificada por experimentos numéricos.
Takafumi MIYATA
Fukuoka Institute of Technology
The copyright of the original papers published on this site belongs to IEICE. Unauthorized use of the original or translated papers is prohibited. See IEICE Provisions on Copyright for details.
Copiar
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
Copiar
@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},}
Copiar
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 -