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首页> 外文期刊>Science in China. Series A, Mathematics, physics, astronomy >Some theoretical comparisons of refined Ritz vectors and Ritz vectors
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Some theoretical comparisons of refined Ritz vectors and Ritz vectors

机译：精制Ritz向量和Ritz向量的一些理论比较

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Refined projection methods proposed by the author have received attention internationally. We are concerned with a conventional projection method and its refined counterpart for computing approximations to a simple eigenpair (λ, x) of a large matrix A. Given a subspace W that contains an approximation to x, these two methods compute approximations (μ,x) and (μ,x) to (λ,x), respectively. We establish three results. First, the refined eigenvector approximation or simply the refined Ritz vector x is unique as the deviation of x from W approaches zero if A is simple. Second, in terms of residual norm of the refined approximate eigenpair (μ, x), we derive lower and upper bounds for the sine of the angle between the Ritz vector x and the refined eigenvector approximation x, and we prove that x ≠ x unless x = x. Third, we establish relationships between the residual norm ‖Ax - μx‖ of the conventional methods and the residual norm ‖Ax - μx‖ of the refined methods, and we show that the latter is always smaller than the former if (μ, x) is not an exact eigenpair of A, indicating that the refined projection method is superior to the corresponding conventional counterpart.
机译：作者提出的改进的投影方法已受到国际关注。我们关注的是传统的投影方法及其精巧的方法，用于计算大矩阵A的简单本征对（λ，x）的近似值。给定一个包含x近似值的子空间W，这两种方法都会计算出近似值（μ，x ）和（μ，x）至（λ，x）。我们建立了三个结果。首先，精简的本征向量近似值或精简的Ritz向量x是唯一的，因为如果A很简单，则x与W的偏差接近零。其次，根据细化近似特征对（μ，x）的残差范数，我们得出Ritz向量x与细化特征向量近似x之间的夹角正弦的上下界，并且证明x≠x，除非x = x。第三，我们建立了常规方法的残差范数“ Ax-μx”和精炼方法的残差范数“ Ax-μx”之间的关系，并且证明了如果（μ，x），后者总是小于前者的并非A的精确特征对，这表明改进的投影方法优于相应的常规对应方法。

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