Convergence of the tridiagonal QR algorithm

A note on the convergence theorem of the tridiagonal QR algorithm with Wilkinson’s shift

“…This view was not explicitly described in any literature, as far as the authors know. However, in actual fact, this view was first implied by Hoffmann-Parlett [1], and then followed by other researchers [7,8,11] (for more detail, see Sects. 2 and 3).…”

“…As stated above, [7,8] showed that, if lim n→∞ β Recall that λ k is the closest eigenvalue to λ l . Let λ j be the second closest one.…”

“…In most cases, the limiting matrix satisfies (11), and thus the actual convergence speed is better than cubic [1,7,8,11]. In the case (11), Parlett demonstrates the asymptotic ratio |β…”

“…The possibility of the loss of cubic convergence, i.e., the occurrence of (12), was discussed in several studies (see [1,7,8,11]). For example, Parlett [7] and Wang [8] show that, in the case (12), the limiting pattern of the lower right 3-by-3 matrix is…”

Convergence of the unitary $QR$ algorithm with a unimodular Wilkinson shift

“…(k) e n can be accepted as an (approximate) eigenvalue of U and the computation continues with the submatrix obtained by deleting the last row and column; sequentially all the eigenvalues are computed and come out in turn [11], [6], [8] …”