計算最大特徵空間之結構可調控二次方法

Abstract

本研究結合冪次法(power method)與源自結構保持倍增演算法(Structure-Preserving Doubling Algo- rithm, SDA)之倍增策略，以加速矩陣運算。儘管SDA 原為求解離散時間代數Riccati 方程(DARE)所 設計，其迭代框架在本研究中被改造應用於主特徵空間之提取。透過變形之倍增程序，可近似求得對應 於主特徵值之特徵向量。然而，對任意矩陣而言，其收斂性未必得以保證。為改善此一問題，本文引入 一種受冪次法啟發之正規化策略:於每次迭代中以矩陣之範數進行縮放，以增強收斂行為。因此，我們 提出一種混合方法，稱為倍增演算法(doubling algorithm)。另方面，為保留有利於收斂之矩陣結構， 本文進一步提出自適應區塊分割演算法(adaptive partition algorithm)，透過適當區塊分割策略，在無需 正規化情況下，仍可保證迭代過程之收斂性。上述兩種方法提供彈性且高效率之主特徵空間計算策略， 並有助於提升迭代型矩陣演算法之整體穩定性。

In this study, we aim to accelerate matrix computations by integrating the power method with a doubling strategy originally developed in the structure-preserving doubling algorithm (SDA). While SDA was initially proposed for solving discrete-time algebraic Riccati equations (DAREs), we adapt its iterative framework to efficiently extract the dominant eigenspace of a given matrix. By applying a variant of the doubling process, we obtain approximations to the eigenvector corresponding to the dominant eigenvalue. However, convergence is not guaranteed for arbitrary matrices. To address this issue, we incorporate a normalization strategy inspired by the power method—scaling the matrix at each iteration by its norm—to enhance convergence behavior. As a result, we propose a hybrid approach, referred to as the doubling algorithm. Alternatively, by appropriately partitioning the matrix to preserve the structure responsible for convergence, we introduce a second method: the adaptive partition algorithm. In this approach, we apply the doubling algorithm without normalization, and show that, under a suitable block-partitioning strategy, the convergence of the iterative process can still be guaranteed. These two methods provide flexible and efficient strategies for computing dominant eigenspaces and improving the overall stability of iterative matrix algorithms.