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Description
We consider the eigenvalue problem $Ax^{(i)} = \lambda_i x^{(i)}$ for a real symmetric matrix $A \in \mathbb{R}^{n \times n}$, where $\lambda_i \in \mathbb{R}$ is an eigenvalue of $A$ and $x^{(i)} \in \mathbb{R}^n$ is the corresponding eigenvector. This work investigates iterative refinement methods to improve the accuracy of eigenvectors $x^{(i)}$.
Efficient methods are known for improving the accuracy of either a single approximate eigenvector or all approximate eigenvectors simultaneously. In practical applications, however, it is often necessary to improve only a subset of eigenvectors: $X = \left( x^{(p_1)}, x^{(p_2)}, \dots, x^{(p_k)} \right) \in \mathbb{R}^{n \times k}, \{p_i\} = \{1, \dots, n\}, 1 \le k \le n$.
The proposed method improves the accuracy of an approximation $\widehat{X}$ to $X$. Given the matrix $A$ and an approximate eigenvector matrix $\widehat{X}$, the goal is to compute $\widetilde{X}$ such that
$\|X - \widetilde{X}\| < \|X - \widehat{X}\|$. A necessary condition for improving accuracy is $\min_{1 \le i \le k} |\lambda_{p_i}| > \max_{k+1 \le j \le n} |\lambda_{p_j}|$.
The proposed method has two main advantages: its computational cost is dominated by matrix multiplications, and it can handle large sparse matrices, making it applicable to mixed-precision and high-precision numerical computations. We report both analytical results and numerical experiments.