Rayleigh Quotient Defender

• Rayleigh Quotient Defender is an advanced iterative method that uses a complex projection to shift non-target eigenvalues into the complex plane for isolation.
• It preserves local cubic convergence by introducing a controlled, complex-valued perturbation, thereby enhancing stability in tight spectral clusters.
• The method is effective in applications like computational quantum mechanics and photonics, where precise interior eigenvalue computation is critical.

The Rayleigh Quotient Defender denotes a complex-projected Rayleigh‐Quotient Iteration (PRQI) scheme specifically constructed to address the notorious unreliability of the classic Rayleigh Quotient Iteration (RQI) when targeting interior eigenvalues of Hermitian matrices in the presence of tightly clustered spectra. The primary innovation is the introduction of a complex-valued projection that leverages the current eigenvector approximation to lift all non-target eigenvalues into the complex plane, thereby isolating the target eigenvalue near the real axis. This approach both preserves the desirable local cubic convergence of RQI and dramatically enhances global robustness, allowing reliable convergence to the intended eigenpair for a much broader range of initial vectors, including scenarios where the classic method is prone to failure (Friess et al., 2023).

1. Mathematical Framework and Notation

Let $A \in \mathbb{C}^{n \times n}$ be Hermitian ($A^* = A$), and let the goal be computation of a simple real eigenpair $(\lambda, v)$ with $\|v\| = 1$ and $A v = \lambda v$. At iteration $k$, the algorithm maintains a unit-norm approximation $v_k$ and computes its Rayleigh quotient

$\rho_k = R_A(v_k) = v_k^* A v_k \in \mathbb{R}.$

The target eigenvalue $\lambda$ is unknown, and $v_k$ should converge to the corresponding eigenvector $v$.

2. Complex Projection Mechanism

The central mechanism is to modify $A$ at each iteration via a complex-valued projection that depends on $v_k$. The rank-$(n-1)$ projector onto $v_k^\perp$ is defined as

$P_k = I - v_k v_k^*,$

and a small real parameter $\alpha_k > 0$ is selected. The perturbed matrix is constructed as

$A_k^{(c)} = A + i \alpha_k P_k.$

Heuristically, for eigenvector $\approx v_k$, the associated eigenvalue $\lambda$ remains near the real axis, while other eigenvalues $\mu \ne \lambda$ are shifted to $\mu + i\alpha_k$. This configuration artificially enlarges the spectral gap between the target and other eigenvalues, increasing the likelihood of convergence to the correct eigenpair even in closely spaced spectra.

3. Rayleigh Quotient Defender Iteration

The PRQI iteration proceeds as follows, with each step leveraging the complex-projected matrix:

1. Form the Rayleigh quotient $\rho_k = v_k^* A v_k$.
2. Select the perturbation $\alpha_k$.
3. Solve the linear system

$(A - (\rho_k - i \alpha_k)I) w_{k+1} = v_k.$

1. Normalize $w_{k+1}$ to obtain $v_{k+1} = w_{k+1} / \|w_{k+1}\|$.
2. Compute the residual $$r_{k+1} = A v_{k+1} - (v_{k+1}^* A v_{k+1}) v_{k+1}$$.
3. Check convergence: $\|r_{k+1}\| < \text{tol}$.

The process repeats until the desired tolerance is achieved. Detailed pseudocode is provided for algorithmic clarity:

Step	Operation	Description
3	$\rho \gets v_k^* A v_k$	Rayleigh quotient computation
4	$\alpha \gets \text{select}_\alpha(v_k,A,\ldots)$	Choice of perturbation (see Section 6 for strategies)
5	Solve $(A-(\rho - i\alpha)I)w=v_k$	Linear system with complex shift
6	$v_{k+1} \gets w / \|w\|$	Normalization
7	$r \gets A v_{k+1}-(v_{k+1}^*A v_{k+1})v_{k+1}$	Residual evaluation

4. Convergence Theory

The Rayleigh Quotient Defender retains local cubic convergence, under suitable parameterization, similar to classic RQI for Hermitian matrices. For $A$ with eigenvalues labeled so that $\lambda_1 = \lambda$ (target), and $\lambda_2$ is the next closest eigenvalue, assume the initial vector $v_0$ satisfies $\theta_0 = \angle(v_0, v_1)$, with $\tan \theta_0$ below a small constant $c$. Selecting $\alpha_k$ such that

$$0 < \alpha_k \le C_\gamma \tan^{1+q} \theta(v_k, v_1),\quad q\in \{0,1\}$$

results in the contraction

$$\tan\theta_{k+1} \le \text{const} \cdot \tan^{2+q} \theta_k.$$

Setting $q=1$ recovers local cubic convergence ($\alpha_k \sim \tan^2 \theta_k$); $q=0$ yields quadratic convergence ($\alpha_k \sim \tan \theta_k$). The proof involves bounding $|\lambda_1 - (\rho_k - i\alpha_k)|$ and invoking the angle-contraction theorem for inverse iteration with complex shifts.

Empirical results corroborate an enlarged basin of attraction: "By lifting all other eigenvalues into $\operatorname{Im} z>0$, PRQI prevents 'jumping' to the wrong real eigenvalue even when the original spectral gap is tiny."

5. Parameter Selection and Practical Implementation

Parameter $\alpha_k$ can be tied directly to the Rayleigh residual at each iterate. Two effective automatic choices are:

• Quadratic convergence: $\alpha_k = \|r_k\|$
• Cubic convergence: $\alpha_k = \|r_k\|^2$

Given

$$\|r_k\| \le \text{spread}(A) \cdot \tan \theta(v_k, v_1),$$

these choices satisfy the requirements from the convergence theorem. In practice, the method is essentially parameter-free; an optional fixed lower bound on $\alpha_k$ (e.g., $\alpha_k \leftarrow \max\{\alpha_k, \epsilon\}$) can be imposed to avoid stagnation, but is rarely necessary.

6. Numerical Experiments

A range of numerical examples establish the enhanced global robustness and local efficiency of the Rayleigh Quotient Defender:

• $3\times3$ diagonal matrix ($A=\operatorname{diag}(-1,s,1)$ as $s\to1$): PRQI maintains robust attraction basins where standard RQI's fail.
• Tridiagonal $[1,2,1]$ and Wilkinson matrices: PRQI converges to the correct eigenpair provided the initial angle is small; classic RQI can converge to incorrect pairs.
• 2D-Laplace (5-point stencil) discretization: PRQI demonstrates robustness analogous to simpler cases.
• Sturm–Liouville operator for photonic crystal fibers: Here, the operator exhibits a band-gap structure with tightly clustered interior eigenvalues. A coarse, localized initial vector suffices for PRQI to converge reliably to the desired eigenvalue in the gap, while RQI often converges to spurious or band‐edge modes in the absence of careful initialization.

These tests collectively demonstrate the method’s invariance to spectral crowding and underline its local cubic speed and significantly more predictable global behavior (Friess et al., 2023).

7. Implications and Context

The Rayleigh Quotient Defender provides a systematic and theoretically principled method for interior eigenvalue targeting in Hermitian eigenproblems with clustered spectra. Its mechanism of selectively "lifting" non-target eigenvalues is notable for increasing the effective gap around the target, mitigating a principal shortcoming of unmodified RQI. A plausible implication is that this family of complex-projected iterations can be adapted for broader classes of eigenproblems where traditional RQI struggles, particularly in high-dimensional or heavily discretized operator contexts. The approach requires minimal tuning—parameter selection is automatic in practical terms—and exhibits the same best-case local convergence rates as classic RQI. Its performance in applied settings, exemplified by finite element discretizations of Sturm–Liouville operators, suggests utility in computational quantum mechanics and photonics where interior eigenvalues are of physical interest and spectral crowding is the norm (Friess et al., 2023).