Single-Shift QR Iteration

• Single-Shift QR Iteration is a fundamental algorithm that computes eigenvalues of real symmetric tridiagonal matrices by iteratively applying shifted QR steps to reduce off-diagonal entries.
• Its efficiency critically depends on the chosen shift strategy, with methods like the Rayleigh and Wilkinson shifts influencing whether convergence is cubic or quadratic.
• The algorithm’s integration with integrable systems, such as the Toda lattice, underpins its robust framework for effective spectral decomposition and practical eigenvalue computations.

The single-shift QR iteration is a fundamental algorithm for computing eigenvalues of real symmetric tridiagonal matrices. This method applies iterated similarity transformations known as shifted QR steps, which are parameterized by a scalar shift and are designed to accelerate convergence of off-diagonal elements to zero—thereby facilitating deflation and ultimately revealing the spectral decomposition of the matrix. The effectiveness and dynamics of the single-shift QR iteration are deeply influenced by the strategy used to select the shift, the algebraic structure of the matrix spectrum, and connections to integrable systems such as the Toda lattice (1108.6030).

1. Definition and Basic Algorithmic Structure

Given a real symmetric tridiagonal matrix $T \in \mathbb{R}^{n \times n}$, the single-shift QR iteration proceeds as follows:

1. Shift selection: Choose a shift $s \in \mathbb{R}$ by a shift strategy $\sigma : {\cal T}_\Lambda \to \mathbb{R}$, where ${\cal T}_\Lambda$ is the isospectral manifold of symmetric tridiagonal matrices similar to a fixed diagonal matrix $\Lambda$.
2. QR factorization: Compute the unique QR factorization (with suitable conventions), $T - sI = Q R$, where $Q$ is orthogonal and $R$ is upper triangular with positive diagonal entries.
3. Similarity transformation: Form $T^{+} = Q^T T Q$.

This produces a new tridiagonal symmetric matrix $T^+$ with the same eigenvalues as $T$ but with typically reduced off-diagonal entries. The iteration is repeated until at least one off-diagonal entry becomes sufficiently small (deflation), at which point the matrix is partitioned and the process recurses on the smaller submatrix.

A signed variant is often used, in which $T - sI = Q_\star R_\star$, with $Q_\star \in SO(n)$ and $R_\star$ having positive entries except possibly for the last. The shifted QR map is then $F_s(T) = Q_\star^T T Q_\star$, which provides improved behavior near deflation.

2. Shift Strategies and Their Impact

The convergence speed and stability of the single-shift QR iteration are highly dependent on the chosen shift strategy:

• Rayleigh shift: $s = (T)_{nn}$, using the lower-right diagonal entry. This strategy is continuous and results in cubic convergence of the bottom off-diagonal entry $b(T) = (T)_{n,n-1}$ away from singularities.
• Wilkinson’s shift: Chooses $s$ as the eigenvalue of the bottom $2 \times 2$ block closest to $(T)_{nn}$. This strategy is discontinuous, with singularities where both eigenvalues are equidistant from $(T)_{nn}$.

Not all shift strategies guarantee cubic convergence at all points. Discontinuous strategies like Wilkinson’s can result in stalling near their singular sets, where only quadratic convergence is observed (1108.6030).

Table: Common Shift Strategies in Single-Shift QR

Strategy	Shift $s$ Definition	Continuity
Rayleigh	$s = (T)_{nn}$	Continuous
Wilkinson	Closest eigenvalue of trailing $2 \times 2$ block to $(T)_{nn}$	Discontinuous

3. Convergence and Deflation Dynamics

The primary goal of shift selection is to accelerate decay of off-diagonal entries, especially $b(T)$. Typically, one monitors $|b(T)|$ and declares deflation when it is sufficiently small. Away from singular sets of the shift strategy, the convergence rate is typically cubic:

$|b(F_\sigma(T))| \leq C|b(T)|^3$

where $C$ is a constant. However, when iterations pass near non-smooth points of the shift strategy, convergence of $b(T)$ may drop to quadratic:

$|b(F_\sigma(T))| \sim O(|b(T)|^2)$

This phenomenon is particularly pronounced near singular supports of discontinuous shift strategies and in the presence of certain structural features in the spectrum (1108.6030).

4. Influence of Spectral Structure: Arithmetic Progressions

The spectral properties of $T$ crucially affect convergence. If the spectrum $\Lambda = \text{diag}(\lambda_1, \dots, \lambda_n)$ is a.p. free (no three eigenvalues in arithmetic progression), then cubic convergence is observed almost everywhere. When an arithmetic progression (especially of three consecutive eigenvalues) is present, resonance effects may force convergence to be strictly quadratic in certain cases.

The Taylor expansion of $b \circ \Phi_s$ near a deflation point highlights that, due to symmetries, the leading order term is cubic under generic conditions. In the presence of arithmetic progressions or near singularities of discontinuous shift strategies, the expansion may yield a quadratic leading term, explaining the observed slowdowns (1108.6030).

5. Integrable Systems Perspective: Toda Lattice and Height Functions

The single-shift QR iteration is intimately linked with the theory of integrable systems. Specifically:

• Toda lattice connection: The QR step acts as the time-1 map of a flow, with complete integrability implying commute relations:

$F_{s_0} \circ F_{s_1} = F_{s_1} \circ F_{s_0}$

• Coordinates near deflation: In a neighborhood of a deflation set (where the bottom off-diagonal entry vanishes), the shifted QR step contracts the $b$-direction, facilitating deflation.
• Height (Lyapunov) functions: There exist real-valued functions $H(T)$ that strictly increase under iteration (except at fixed points), providing a tool to control the number and location of "bad" (quadratic) steps and to establish uniform bounds on how long the algorithm can remain near non-smooth regions.

This integrable systems perspective explains the robust deflation mechanism and provides a geometric interpretation of the convergence phenomena (1108.6030).

6. Practical Implementation and Efficiency Enhancements

The standard reduction of a general symmetric matrix to tridiagonal form is always possible by orthogonal similarity, and this is the preferred input for single-shift QR algorithms.

Computational workflow:

1. Orthogonally reduce the matrix to symmetric tridiagonal form.
2. At each step, select a shift (using a chosen strategy).
3. Factor $T - sI = Q R$ (or the signed variant).
4. Update via $T^{+} = Q^T T Q$.
5. Monitor $|b(T)|$, deflate when sufficiently small, and recurse as necessary.

Algorithmic refinements:

• Using signed QR steps smooths behavior near deflation.
• Exploiting a well-chosen discontinuous shift (such as Wilkinson) enables rapid, systematic bottom deflation in practice, often outperforming smooth strategies.
• The convergence rate is typically cubic, with rare quadratic slowdowns.
• Height functions can, in principle, be used to monitor and guarantee overall progress.

7. Conclusion

Single-shift QR iteration is a spectrum-preserving, deflation-focused algorithm for tridiagonal symmetric matrices, underpinned by a blend of classical numerical linear algebra and integrable systems theory. Its practical convergence rate depends on the interplay between shift strategy smoothness and spectral structure. The universally observed cubic rate, barring exceptional steps dictated by the presence of arithmetic progressions or non-smoothness in the shift strategy, underscores its effectiveness for eigenvalue computations. Discontinuous shift strategies such as Wilkinson's—despite local quadratic slowdowns—often facilitate rapid and robust deflation, which explains their prevalence in numerical linear algebra software (1108.6030).