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：本文提出了一种新的投影瑞利商迭代法，旨在通过在每个迭代步中融入目标特征向量的近似信息，以改进经典瑞利商迭代的收敛特性。对于厄米矩阵，经典RQI虽具有局部三次收敛性，但其全局行为可能难以预测，即使从精确的初始条件出发，也可能收敛到远离目标的特征值。当特征值间隔紧密时，这一问题尤为突出。新算法的核心思想是在每一步对原矩阵施加一个复值投影（该投影依赖于当前的特征向量近似），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谱中其余部分之间的间隔。通过更有效地利用特征向量近似，新方法具有更稳健的收敛行为，且相比经典RQI，能在更广泛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，并提供了若干数值算例以展示其改进的全局收敛性能。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算子的带隙谱中的特征值，其中目标与非目标特征值间隔紧密。算例表明，即使在特征值间距极小时，新方法仍能收敛至所需特征对，且常在经典RQI失效时取得成功。