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

Abstract

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 question whether the rate of convergence of the algorithm is always cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let \(T_{ {\mathcal {X}}}\) be the 3×3 matrix having only two nonzero entries \((T_{ {\mathcal {X}}})_{12}=(T_{ {\mathcal {X}}})_{21}=1\) and let \({\mathcal {T}}_{\varLambda }\) be the set of real, symmetric tridiagonal matrices with the same spectrum as \(T_{ {\mathcal {X}}}\) . There exists a neighborhood \(\boldsymbol {{\mathcal {U}}}\subset {\mathcal {T}}_{\varLambda }\) of \(T_{ {\mathcal {X}}}\) which is invariant under Wilkinson’s shift strategy with the following properties. For \(T_{0}\in \boldsymbol {{\mathcal {U}}}\) , the sequence of iterates (T k ) exhibits either strictly quadratic or strictly cubic convergence to zero of the entry (T k )23. In fact, quadratic convergence occurs exactly when \(\lim T_{k}=T_{ {\mathcal {X}}}\) . Let \(\boldsymbol {{\mathcal {X}}}\) be the union of such quadratically convergent sequences (T k ): The set \(\boldsymbol {{\mathcal {X}}}\) has Hausdorff dimension 1 and is a union of disjoint arcs \(\boldsymbol {{\mathcal {X}}}^{\sigma}\) meeting at \(T_{ {\mathcal {X}}}\) , where σ ranges over a Cantor set.