论文标题
矩阵特征值问题的理论和可计算的最佳子空间扩展
Theoretical and Computable Optimal Subspace Expansions for Matrix Eigenvalue Problems
论文作者
论文摘要
考虑矩阵特征值问题的最佳子空间扩展问题$ ax =λx$：哪个向量$ w $在当前子空间$ \ mathcal {v} $（乘以$ a $）后，提供了最佳的子空间扩展，以扩展与所需的$ x $ x $ x $ x $相比， $ \ MATHCAL {V} _W = \ MATHCAL {V}+{\ rm span} \ {aw \} $，I.E.这个问题很重要，因为许多迭代方法构建了嵌套子空间，这些子空间将$ \ MATHCAL {V} $连续地扩展到$ \ Mathcal {V} _W $。 ye（线性代数应用程序，428（2008），第911--918页）的$ w_ {opt} $的表达式，但无法利用它来构建一个可计算的（几乎）最佳扩展的子空间。他转向得出$ \ cos \ Angle（\ Mathcal {v} _W，x）$的最大表征，用于{\ em给出} $ w \ in \ Mathcal {v} $时，我们将您的最大化表征概括为一般情况，并找到其最大化器。我们的主要贡献包括$ w_ {opt} $，$（i-p_v）aw_ {opt} $的明确表达方式以及最佳扩展的子空间$ \ nathcal {v} _ {w_ {w_ {opt}} $ $ a $ a $ a $ a $ a $ a $ a $ p_v $，其中$ p_v $是Orthogonal Projector to Orthogonal Projector to $ Maths $ \ Mathcal} $ {这些结果被充分利用，以在标准，谐波，精制和精制的谐波雷利 - 里兹方法的框架内获得可计算的最佳扩展子空间。我们展示了如何有效地实施所提出的子空间扩展方法。数值实验证明了我们可计算的最佳扩展的有效性。
Consider the optimal subspace expansion problem for the matrix eigenvalue problem $Ax=λx$: Which vector $w$ in the current subspace $\mathcal{V}$, after multiplied by $A$, provides an optimal subspace expansion for approximating a desired eigenvector $x$ in the sense that $x$ has the smallest angle with the expanded subspace $\mathcal{V}_w=\mathcal{V}+{\rm span}\{Aw\}$, i.e., $w_{opt}=\arg\max_{w\in\mathcal{V}}\cos\angle(\mathcal{V}_w,x)$? This problem is important as many iterative methods construct nested subspaces that successively expand $\mathcal{V}$ to $\mathcal{V}_w$. An expression of $w_{opt}$ by Ye (Linear Algebra Appl., 428 (2008), pp. 911--918) for $A$ general, but it could not be exploited to construct a computable (nearly) optimally expanded subspace. He turns to deriving a maximization characterization of $\cos\angle(\mathcal{V}_w,x)$ for a {\em given} $w\in \mathcal{V}$ when $A$ is Hermitian. We generalize Ye's maximization characterization to the general case and find its maximizer. Our main contributions consist of explicit expressions of $w_{opt}$, $(I-P_V)Aw_{opt}$ and the optimally expanded subspace $\mathcal{V}_{w_{opt}}$ for $A$ general, where $P_V$ is the orthogonal projector onto $\mathcal{V}$. These results are fully exploited to obtain computable optimally expanded subspaces within the framework of the standard, harmonic, refined, and refined harmonic Rayleigh--Ritz methods. We show how to efficiently implement the proposed subspace expansion approaches. Numerical experiments demonstrate the effectiveness of our computable optimal expansions.
