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## Orthogonal Deflating Transformation

We shall utilize a special orthogonal transformation to implement the deflation schemes mentioned above. The deflation schemes are related to an eigenvector associated with a Ritz value that is to be deflated (either locked or purged). Given a vector of unit length, the algorithm shown in Algorithm 4.9 computes an orthogonal matrix such that (hence ). This orthogonal matrix has a very special form and may be written as Q = R + y e_1^*, with R e_1 = 0 , R^* y = 0, where is upper triangular. It may also be written as Q = L + y g^* , with L e_1 = 0 , L^* y = e_1 - g, where is lower triangular and . Here we assume that .

Now, consider the matrix . The substitutions , from (4.24) and (4.23) and the facts and will give Q^* T Q &=& Q^* T ( R + y e_1^*)
&=& (L^* + g y^*) T R + e_1 e_1^*
&=& L^*TR + g y^*T R + e_1 e_1^*. Since both and are upper triangular, it follows that is upper Hessenberg with the first row and the first column each being zero due to . Also, . From this we conclude that must also be symmetric and hence tridiagonal. Therefore, we see that is of the form where is symmetric and tridiagonal.

It should be noted that, as computed by Algorithm 4.9, will have componentwise relative errors on the order of machine precision with no element growth. Subsections     Next: Locking or Purging a Up: Implicitly Restarted Lanczos Method Previous: Deflation and Stopping Rules   Contents   Index
Susan Blackford 2000-11-20