The restarted QR-algorithm for eigenvalue computation of structured matrices

The QR-algorithm is a popular numerical method for the computation of eigenvalues of matrices. All eigenvalues of a general n×n upper Hessenberg matrix typically can be computed in O(n3) arithmetic floating point operations using O(n2) storage locations. When the upper Hessenberg matrix is Hermitian or unitary, then it can be represented by O(n) parameters, and there are variants of the QR-algorithm that reduce the operation count for computing all eigenvalues to O(n2) arithmetic floating point operations and the storage requirement to O(n) locations. However, for many structured matrices that can be represented with O(n) storage locations, available implementations of the QR-algorithm require O(n3) arithmetic floating point operations and O(n2) storage locations to determine all eigenvalues. This note shows that for some of the latter matrices, the operation count can be reduced to O(n2) arithmetic floating point operations and the memory requirement to O(n) storage locations by periodically restarting the QR-algorithm.