Abstract
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 (‚;x) of a large matrix A. Given a subspace W that contains an approximation to x, these two methods compute approximations ("; ˜) and ("; ˆ x) to (‚;x), respectively. We establish three results. First, the refined eigenvector approximation or simply the refined Ritz vectoris unique as the deviation of x from W approaches zero if ‚ is simple. Second, in terms of residual norm of the refined approximate eigenpair ("; ˆ), we derive lower and upper bounds for the sine of the angle between the Ritz vector ˜ x and the refined eigenvector approximation ˆ x, and we prove that ˜ x 6 ˆ unless ˆ x = x. Third, we establish relationships between the residual norm kA˜ x i "˜ xk of the conventional methods and the residual norm kAˆ i "ˆ xk of the refined methods, and we show that the latter is always smaller than the former if ("; ˆ) is not an exact eigenpair of A, indicating that the refined projection method is superior to the corresponding conventional counterpart.
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