子空间迭代的逐元素收敛行为
Element-wise Convergence Behavior of Subspace Iteration
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中文总结 AI 辅助
本文研究子空间迭代的逐元素收敛行为，针对对角矩阵情形推导了迭代矩阵和Ritz向量各元素模的精确渐近衰减速率，并推广至实对称或复正规矩阵。
中文摘要 AI 辅助
本文研究子空间迭代的逐元素收敛行为。所有结果均针对 \\(\mathbb{F}=\mathbb{R}\\) 和 \\(\mathbb{F}=\mathbb{C}\\) 两种情况建立。对于对角矩阵 \\(\Lambda=\mathrm{diag}(\lambda_1,\ldots,\lambda_n)\\)，其中 \\(|\lambda_1|>\cdots>|\lambda_n|>0\\)，我们分析了由子空间迭代生成的迭代矩阵序列中每个元素的模，分别考虑不采用和采用 Rayleigh--Ritz 过程的情形。在对初始矩阵 \\(X\in\mathbb{F}^{n\times m}\\) 的温和假设下，我们首先推导了在不采用 Rayleigh--Ritz 过程的子空间迭代中 \\(|Q_k(i,j)|\\) 的精确渐近表达式：当 \\(i\neq j\\) 时，元素以 \\((|\lambda_{\max\{i,j\}}|/|\lambda_{\min\{i,j\}}|)^k\\) 的速率衰减；而 \\(|Q_k(j,j)|\\) 与 1 的偏差以 \\(\max\{ |\lambda_j|/|\lambda_{j-1}|, |\lambda_{j+1}|/|\lambda_j| \}^{2k}\\) 的速率衰减，其中显式系数由 \\(X\\) 的 LU 分解确定。对于 Rayleigh--Ritz 变体，我们获得了归一化 Ritz 向量 \\(Z_k\\) 的逐元素界。具体而言，对于 \\(|Z_k(i,j)|\\)，当 \\(i\geq m+1\\) 时的非对角元素以 \\((|\lambda_i|/|\lambda_j|)^k\\) 的速率衰减，当 \\(i\leq m\\) 时的非对角元素以 \\((|\lambda_{m+1}|^2/(|\lambda_i||\lambda_j|))^k\\) 的速率衰减，而 \\(|Z_k(j,j)|\\) 与 1 的偏差以 \\((|\lambda_{m+1}|/|\lambda_j|)^{2k}\\) 的速率衰减。这些结果给出了子空间迭代收敛行为的逐元素描述。在正交或酉基变换之后，这些结果适用于实对称或复正规矩阵。
英文摘要
This paper studies the element-wise convergence behavior of subspace iteration. All results are established for both \(\mathbb{F}=\mathbb{R}\) and \(\mathbb{F}=\mathbb{C}\). For a diagonal matrix \(Λ=\mathrm{diag}(λ_1,\ldots,λ_n)\) with \(|λ_1|>\cdots>|λ_n|>0\), we analyze the modulus of each entry of the iterative matrix sequence generated by subspace iteration, both without and with the Rayleigh--Ritz procedure. Under mild assumptions on the initial matrix \(X\in\mathbb{F}^{n\times m}\), we first derive exact asymptotic expressions for \(|Q_k(i,j)|\) in the subspace iteration without the Rayleigh--Ritz procedure: entries with \(i\neq j\) decay as \((|λ_{\max\{i,j\}}|/|λ_{\min\{i,j\}}|)^k\), and the deviation of \(|Q_k(j,j)|\) from \(1\) decays as \(\max\{|λ_j|/|λ_{j-1}|,|λ_{j+1}|/|λ_j|\}^{2k}\), with explicit coefficients determined by the LU factorization of \(X\). For the Rayleigh--Ritz variant, we obtain element-wise bounds for the normalized Ritz vectors \(Z_k\). Specifically, for \(|Z_k(i,j)|\), off-diagonal entries with \(i\geq m+1\) decay as \((|λ_i|/|λ_j|)^k\), off-diagonal entries with \(i\leq m\) decay as \((|λ_{m+1}|^2/(|λ_i||λ_j|))^k\), and the deviation of \(|Z_k(j,j)|\) from \(1\) decays as \((|λ_{m+1}|/|λ_j|)^{2k}\). These results give an element-wise description of the convergence behavior of subspace iteration. After an orthogonal or unitary change of basis, the results apply to real symmetric or complex normal matrices.
