The Asymptotics of Wilkinson's Shift: Loss of Cubic Convergence

One of the most widely used methods for eigenvalue computation is the QR iteration with Wilkinson’s shift: here the shift s is the eigenvalue of the bottom 2 × 2 principal minor closest to the corner entry. It has been a long-standing conjecture that the rate of convergence of the algorithm is cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let TX be the 3 × 3 matrix having only two nonzero entries (TX )12 = (TX )21 = 1 and let TΛ be the set of real, symmetric tridiagonal matrices with the same spectrum as TX . There exists a neighborhood U ⊂ TΛ of TX which is invariant under Wilkinson’s shift strategy with the following properties. For T0 ∈ U, the sequence of iterates (Tk) exhibits either strictly quadratic or strictly cubic convergence to zero of the entry (Tk)23. In fact, quadratic convergence occurs exactly when lim Tk = TX . Let X be the union of such quadratically convergent sequences (Tk)...

• Type: Journal
• Year: 2010
• Where: FOCM
• Authors: Ricardo S. Leite, Nicolau C. Saldanha, Carlos Tomei
• Keywords: Convergence, Eigenvalue Computation, FOCM 2010, Principal Minor Closest