This preview shows page 1. Sign up to view the full content.
The Francis Algorithm Recall that we have a shifted QR iteration that converges quickly to a reduced Hessenberg matrix if the s i are close to an eigenvalue of H (in fact, only one iteration does it if s i is an eigenvalue of H (this is called an ultimate shift )). If H is reduced, we can decouple (or deﬂate) and continue with a strictly smaller problem than before. Here is the iteration: Q i R i = H i - s i I H i +1 = R i Q i + s i I = Q t i H i Q i Nonsymmetric real matrices may have complex eigenvalues, which must occur in conjugate pairs: u + iv and u - iv . If we apply a complex shift s i = u + iv to the iteration, then Q i , R i and H i +1 will be complex. This requires more storage, and more computation (one complex multiplication requires 4 real multiplications and 2 real additions). Now if we immediately follow the s i = u + iv shift iteration with an iteration with shift s i +1 = u - iv , then everything becomes again real. We can understand this by noting that two iterations above gives the same
This is the end of the preview. Sign up to access the rest of the document.
This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.
- Fall '10