We introduce a new Projected Rayleigh Quotient Iteration aimed at improving the convergence behaviour of classic Rayleigh Quotient iteration (RQI) by incorporating approximate information about the target eigenvector at each step. While classic RQI exhibits local cubic convergence for Hermitian matrices, its global behaviour can be unpredictable, whereby it may converge to an eigenvalue far away from the target, even when started with accurate initial conditions. This problem is exacerbated when the eigenvalues are closely spaced. The key idea of the new algorithm is at each step to add a complex-valued projection to the original matrix (that depends on the current eigenvector approximation), such that the unwanted eigenvalues are lifted into the complex plane while the target stays close to the real line, thereby increasing the spacing between the target eigenvalue and the rest of the spectrum. Making better use of the eigenvector approximation leads to more robust convergence behaviour and the new method converges reliably to the correct target eigenpair for a significantly wider range of initial vectors than does classic RQI. We prove that the method converges locally cubically and we present several numerical examples demonstrating the improved global convergence behaviour. In particular, we apply it to compute eigenvalues in a band-gap spectrum of a Sturm-Liouville operator used to model photonic crystal fibres, where the target and unwanted eigenvalues are closely spaced. The examples show that the new method converges to the desired eigenpair even when the eigenvalue spacing is very small, often succeeding when classic RQI fails.

翻译：我们提出了一种新的投影瑞利商迭代方法，旨在通过每一步引入目标特征向量的近似信息，改善经典瑞利商迭代（RQI）的收敛性能。经典RQI对Hermitian矩阵具有局部三次收敛性，但其全局行为可能不可预测——即便初始条件精确，也可能收敛到远离目标的特征值。当特征值间隔较小时，该问题会进一步加剧。新算法的核心思想是在每一步向原始矩阵添加一个复值投影（该投影依赖于当前特征向量近似），使得非目标特征值被提升至复平面，而目标特征值仍保持在实轴附近，从而增大目标特征值与其余谱之间的间隔。通过更充分地利用特征向量近似信息，新方法展现出更稳健的收敛行为，能够在远宽于经典RQI的初始向量范围内可靠地收敛至正确的目标特征对。我们证明了该方法具有局部三次收敛性，并通过多个数值算例展示了其改进的全局收敛性能。特别地，我们将该方法应用于计算Sturm-Liouville算子的带隙谱中的特征值（该算子用于建模光子晶体光纤），其中目标特征值与干扰特征值间隔极小。算例表明，即便在特征值间隔极小时，新方法仍能收敛到期望的特征对，且往往在经典RQI失效时成功完成收敛。