A complex-projected Rayleigh quotient iteration for targeting interior eigenvalues

Abstract

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.

Figures (6)

• Figure 1: Section of the spectrum of a matrix $A$ (orange circles) and the perturbed matrix $\widetilde{A}$ (blue triangles). Most of the perturbed eigenvalues $\widetilde{\lambda}$ have $\text{Im}({\widetilde{\lambda}}) \approx 1$ while one eigenvalue lies close to the real line. The matrix $A$ and the vector $\bm{u}$ that were used to construct $\widetilde{A}$ for this plot come from Example \ref{['ex:sturm']} in Section \ref{['sec:numerics']}.
• Figure 2: Visualisation of the basins of attraction for classic RQI (top row) and PRQI (bottom row) for different eigenvalue gaps in Example \ref{['ex:comparison3x3']}. If the eigenvalue gap is small (right column), the borders between the basins of attraction deteriorate for classic RQI; for PRQI the border is more regular.
• Figure 3: Plot of the angle between the initial vector and the target eigenvector against the computed eigenvalue for the four test matrices in Example \ref{['ex:comparison2']}. The target eigenvalue is depicted by the solid black line. While classic RQI sometimes fails even for very small angles (i.e., very accurate initial vectors), PRQI consistently produces the correct result as soon as the angle is sufficiently small.
• Figure 4: Plots of one of the eigenfunctions corresponding to a target eigenvalue (left) and of the eigenfunction corresponding to an unwanted, spurious eigenvalue (right) for the Sturm-Liouville problem in \ref{['eq:schroedinger']}.
• Figure 5: Part of the spectrum of the original and the perturbed Sturm--Liouville eigenvalue problem \ref{['eq:schroedinger']}. The target eigenvalue in the original spectrum (red cross) is very close to the unwanted eigenvalues in the band (hatched region) and the spurious eigenvalue (bordered circle). The perturbed eigenvalues (blue triangles) are raised into the complex plane and only the target eigenvalue (blue cross) stays near the real line.

Theorems & Definitions (17)

• Theorem 2.1: boermmehl, Theorem 4.10
• Definition 2.2: Rayleigh Quotient
• Theorem 2.3: boermmehl, Theorem 4.6
• Proposition 2.4: saad2011, Cor. 3.4 and Thm. 3.9
• Theorem 2.5: boermmehl, Proposition 4.13
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3