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#### Relationship to Symmetric Lanczos.

From equations (6.7), (6.8), and (6.9), we have We note that is symmetric and tridiagonal, since is bidiagonal. Comparing to equation (4.10), we see that Algorithm 6.1 computes the same information as Lanczos (Algorithm 4.6) applied to the Hermitian matrix . Conversely, if we apply Lanczos to to get a tridiagonal matrix , then can be obtained by taking the upper triangular Cholesky factor of .

Similarly, one gets Again, is symmetric and tridiagonal. So again comparing to equation (4.10), we see that Algorithm 6.1 computes the same information as Lanczos (Algorithm 4.6) applied to the Hermitian matrix . Conversely, if we apply Lanczos to to get a tridiagonal matrix , then can be obtained by taking the upper triangular Cholesky factor of , where is the identity matrix with its columns in reverse order (so that is gotten by reversing the order of the rows and then the columns of ), and then .

Finally, suppose one applies Lanczos (Algorithm 4.6) to with the special starting vector to generate the Lanczos vectors The first step of Algorithm 4.6 yields The next step of Algorithm 4.6 yields Continuing in this fashion, we see that two steps of Algorithm 4.6 compute the same information as one step of Algorithm 6.1: However, Algorithm 4.6 does twice as many matrix-vectors multiplications by and as Algorithm 6.1 (half of them by zero vectors), so that Algorithm 6.1 will generally use half the time and space. Conversely, if Lanczos is applied to to obtain a tridiagonal matrix , then will have zeros on its diagonal, and the off-diagonal entries will be identical to the nonzero entries of (notation from equation (6.4)): Because of these equivalences, all the algorithmic variations and convergence properties of Lanczos from §4.4 apply to Algorithm 6.1.     Next: Software Availability Up: Golub-Kahan-Lanczos Method Previous: Golub-Kahan-Lanczos Bidiagonalization Procedure.   Contents   Index
Susan Blackford 2000-11-20