Francis

# Francis - The Francis Algorithm Recall that we have a shifted QR iteration that converges quickly to a reduced Hessenberg matrix if the si are

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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
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## This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.

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