Some theoretical comparisons of refined Ritz vectors and Ritz vectors
Some theoretical comparisons of refined Ritz vectors and Ritz vectors
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DOI：
10.1360/04za0020
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
Zhongxiao Jia
中科院分区：
文献类型：
--
作者：
Zhongxiao Jia
AbstractRefined 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, ϰ) of a large matrix A. Given a subspace W that contains an approximation to ϰ, these two methods compute approximations (μ $$\tilde x$$ ) and (μ $$\hat x$$ ) to (λ, ϰ), respectively. We establish three results. First, the refined eigenvector approximation or simply the refined Ritz vector $$\hat x$$ is unique as the deviation of ϰ from W approaches zero if λ is simple. Second, in terms of residual norm of the refined approximate eigenpair (μ, $$\hat x$$ ), we derive lower and upper bounds for the sine of the angle between the Ritz vector $$\tilde x$$ and the refined eigenvector approximation $$\hat x$$ , and we prove that $$\tilde x \ne \hat x$$ unless $$\hat x = x$$ . Third, we establish relationships between the residual norm $$\left\| {A\tilde x - \mu \tilde x} \right\|$$ of the conventional methods and the residual norm $$\left\| {A\hat x - \mu \hat x} \right\|$$ of the refined methods, and we show that the latter is always smaller than the former if (μ, $$\hat x$$ ) is not an exact eigenpair of A, indicating that the refined projection method is superior to the corresponding conventional counterpart.
