The paper describes a heuristic algorithm for solving a generalized Hermitian eigenvalue problem fast. The algorithm searches a subspace for an approximate solution of the problem. If the approximate solution is unacceptable, the subspace is expanded to a larger one, and then, in the expanded subspace a possibly better approximated solution is computed. The algorithm iterates these two steps alternately. Thus, the speed of the convergence of the algorithm depends on how to generate a subspace. In this paper, we derive a Riccati equation whose solution can correct the approximate solution of a generalized Hermitian eigenvalue problem to the exact one. In other words, the solution of the eigenvalue problem can be found if a subspace is expanded by the solution of the Riccati equation. This is a feature the existing algorithms such as the Krylov subspace algorithm implemented in the MATLAB and the Jacobi–Davidson algorithm do not have. However, similar to solving the eigenvalue problem, solving the Riccati equation is time-consuming. We consider solving the Riccati equation with low accuracy and use its approximate solution to expand a subspace. The implementation of this heuristic algorithm is discussed so that the computational cost of the algorithm can be saved. Some experimental results show that the heuristic algorithm converges within fewer iterations and thus requires lesser computational time comparing with the existing algorithms.

中文翻译：

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一种求解广义Hermitian特征值问题的Riccati型算法

该论文描述了一种用于快速求解广义 Hermitian 特征值问题的启发式算法。该算法在子空间中搜索问题的近似解。如果近似解不可接受，则将子空间扩展为更大的子空间，然后在扩展的子空间中计算可能更好的近似解。该算法交替迭代这两个步骤。因此，算法收敛的速度取决于如何生成子空间。在本文中，我们导出了一个 Riccati 方程，该方程的解可以将广义 Hermitian 特征值问题的近似解修正为精确解。换句话说，如果通过 Riccati 方程的解来展开子空间，就可以找到特征值问题的解。这是现有算法（例如在 MATLAB 中实现的 Krylov 子空间算法和 Jacobi-Davidson 算法）所不具备的功能。但是，与求解特征值问题类似，求解 Riccati 方程也很耗时。我们考虑以低精度求解 Riccati 方程，并使用其近似解来扩展子空间。讨论了这种启发式算法的实现，从而可以节省算法的计算成本。一些实验结果表明，启发式算法在更少的迭代中收敛，因此与现有算法相比需要更少的计算时间。我们考虑以低精度求解 Riccati 方程，并使用其近似解来扩展子空间。讨论了这种启发式算法的实现，从而可以节省算法的计算成本。一些实验结果表明，启发式算法在更少的迭代中收敛，因此与现有算法相比需要更少的计算时间。我们考虑以低精度求解 Riccati 方程，并使用其近似解来扩展子空间。讨论了这种启发式算法的实现，从而可以节省算法的计算成本。一些实验结果表明，启发式算法在更少的迭代中收敛，因此与现有算法相比需要更少的计算时间。