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法，旨在通过在每个步骤中融入关于目标特征向量的近似信息，改善经典瑞利商迭代（RQI）的收敛行为。经典RQI对于厄米矩阵具有局部三次收敛性，但其全局行为可能难以预测——即使从精确初始条件出发，也可能收敛到远离目标的特征值。当特征值间隔较小时，这一问题尤为突出。新算法的核心思想是：在每一步中，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值），使得非目标特征值被提升到复平面，而目标特征值保持接近实轴，从而增大目标特征值与其余谱之间的间隔。更有效地利用特征向量近似值能够带来更稳健的收敛行为，且与经典RQI相比，新方法在更广泛的初始向量范围内能可靠地收敛到正确的目标特征对。我们证明了该方法具有局部三次收敛性，并给出多个数值示例以展示其改进后的全局收敛行为。特别地，我们将其应用于计算用于模拟光子晶体光纤的Sturm-Liouville算子带隙谱中的特征值，其中目标特征值与干扰特征值间隔极小。示例表明，即使特征值间隔非常小，新方法仍能收敛到目标特征对，且常在经典RQI失效时成功。