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6.4. PROJECTION TO STETTER-STRUCTURE 99
(i) If the norm of the residual r j is smaller than a specified tolerance threshold then
the output of the eigenvalue solver is {u j ,θ j } and θ j is approximately the global
minimum of the polynomial of interest.
(ii) If the norm of the residual r is larger than or equal to the tolerance and
max(Im(u j )) < 1 × 10 −5 then the absolute values of u j are projected to the Stetter
structure as described in the previous section yielding the vector û j . The corresponding
eigenvalue ˆθ j is computed using the Rayleigh quotient as in Equation (6.3).
Finally, this new eigenvalue/eigenvector pair is accepted as a next improved pair if
the norm of the residual decreases or is discarded otherwise:
⎧
⎪⎨ (û j , ˆθ j ) if ∣ ∣ Aû j − ˆθ j û ∣ j ∣ 2
(u j+1 ,θ j+1 2