Solving Eigenproblems: From Arnoldi via Jacobi-Davidson to the Riccati Method
Solving Eigenproblems: From Arnoldi via Jacobi-Davidson to the Riccati Method
复制标题
解决特征问题：从阿诺尔迪到雅可比-戴维森到里卡蒂方法
DOI：
10.1007/3-540-36487-0_18
复制
发表时间：
2002
期刊：
影响因子：
--
通讯作者：
J. Brandts
中科院分区：
文献类型：
--
作者：
J. Brandts
将特征问题表述为广义代数 Riccati 方程消除了特征向量的非唯一性问题。这一基本思想催生了 Sleijpen 和 Van der Vorst (1996) 的 Jacobi-Davidson (JD) 方法。当当前迭代足够接近目标解决方案时，JD 会二次收敛。不幸的是，可能需要付出相当大的努力才能足够接近这个解决方案。在本文中，我们提出了对此的补救措施。我们建议用低维 Riccati 方程代替 Riccati 方程并精确求解，而不是对 Riccati 方程进行线性化（在 JD 中完成）并用低维线性系统替换线性化。由此产生的 Riccati 算法的性能与 JD 相比非常有利，而与 JD 相比，每次迭代的额外成本实际上可以忽略不计。
The formulation of eigenproblems as generalized algebraic Riccati equations removes the non-uniqueness problem of eigenvectors. This basic idea gave birth to the Jacobi-Davidson (JD) method of Sleijpen and Van der Vorst (1996). JD converges quadratically when the current iterate is close enough to the solution that one targets for. Unfortunately, it may take quite some effort to get close enough to this solution. In this paper we present a remedy for this. Instead of linearizing the Riccati equation (which is done in JD) and replacing the linearization by a low-dimensional linear system, we propose to replace the Riccati equation by a low-dimensional Riccati equation and to solve it exactly. The performance of the resulting Riccati algorithm compares extremely favorable to JD while the extra costs per iteration compared to JD are in fact negligible.