A Unifying Framework for Doubling Algorithms

The existing doubling algorithms have been proven efficient for several important nonlinear matrix equations arising from real-world engineering applications. In a nutshell, the algorithms iteratively compute a basis matrix, in one of the two particular forms, for the eigenspace of some matrix pencil associated with its eigenvalues in certain complex region such as the left-half plane or the open unit disk, and their success critically depends on that the interested eigenspace do have a basis matrix taking one of the two particular forms. However, that requirement in general cannot be guaranteed. In this paper, a new doubling algorithm, called the $Q$-doubling algorithm, is proposed. It includes the existing doubling algorithms as special cases and does not require that the basis matrix takes one of the particular forms. An application of the $Q$-doubling algorithm to solve eigenvalue problems is investigated with numerical experiments that demonstrate its superior robustness to the existing doubling algorithms.

💡 Research Summary

The paper addresses a fundamental limitation of existing structure‑preserving doubling algorithms (SD‑ASF1 and SD‑ASF2) used for solving nonlinear matrix equations and associated eigenvalue problems. Traditional doubling methods assume that the basis matrix of the invariant subspace can be written in one of two special forms, namely (\begin{bmatrix}I\X\end{bmatrix}) or (\begin{bmatrix}Y\I\end{bmatrix}) (or (\begin{bmatrix}I\Y\end{bmatrix}) for the second standard form). This assumption is often violated in practice because the leading block of the basis matrix may be singular or nearly singular, leading to numerical instability or outright failure of the algorithm.

The authors propose a new “Q‑doubling” algorithm that removes the need for these special basis forms. The key idea is to introduce a permutation‑or‑orthogonal matrix (Q) that dynamically re‑orders and orthogonalizes the basis matrix at each iteration, transforming it into a generalized Q‑standard form (SF(_Q)). In this form the basis can always be expressed as (Q^{!T}Z = \begin{bmatrix}I\X\end{bmatrix}) with a suitably chosen (Q). Theorem 3.1 guarantees the existence of such a (Q) and provides a bound (|X|_2 \le \sqrt{mn+1}), ensuring that the magnitude of the unknown block (X) remains moderate.

The algorithm proceeds in three stages:

1. Initialization – Starting from a regular matrix pencil (A-\lambda B), the method applies a similarity transformation (P) and an initial orthogonal matrix (Q_0) to obtain a pencil in the Q‑standard form. This step mirrors the preprocessing used in classical doubling algorithms (Möbius transformations, scaling, etc.) but adds the extra freedom of choosing (Q_0) to control the size of the sub‑block.

2. Doubling iteration – The classic doubling transformation (Theorem 2.1) is applied unchanged to generate a sequence of pencils ((A_i,B_i)). After each doubling step, the current basis matrix (\tilde Z_i) is recomputed, and a new orthogonal matrix (Q_{i+1}) is selected so that (Q_{i+1}^{!T}\tilde Z_i) again has the desired block‑identity structure. This dynamic update prevents the growth of (|X_i|) and maintains numerical stability even when the original basis is ill‑conditioned.

3. Convergence check – Under the standard spectral conditions (\rho(M)<1) and (\rho(N)<1) (where (M) and (N) are the reduced matrices governing the two invariant subspaces), the algorithm guarantees geometric convergence of the iterates (X_i) and (Y_i) to the true solution. The paper also treats the critical case (\rho(M)\rho(N)=1) with a detailed analysis in Appendix A, showing that the dynamic (Q) updates preserve convergence provided certain Jordan‑block conditions hold.

The authors demonstrate that the Q‑doubling framework subsumes the existing SD‑ASF1 and SD‑ASF2 algorithms as special cases (by fixing (Q) to the identity). Moreover, the new method can be applied directly to the discrete‑time algebraic Riccati equation (DARE), continuous‑time algebraic Riccati equation (CARE), and M‑matrix algebraic Riccati equation (MARE) without any additional reformulation, because the underlying nonlinear equations are unchanged; only the basis handling differs.

Two numerical experiments illustrate the practical benefits:

• Experiment 1 (DARE) – Compared with the classical SD‑ASF1, the Q‑doubling algorithm achieves the same or higher accuracy while remaining stable when the solution matrix (X) is large or when the leading block of the original basis is near singular. The relative error drops to the order of (10^{-8}) after roughly 25 iterations, whereas the traditional method stalls or diverges.

• Experiment 2 (General pencil with spectrum crossing the unit circle) – For a randomly generated pencil whose eigenvalues straddle the unit circle, the conventional algorithms fail to converge because the required basis form cannot be realized. The Q‑doubling algorithm, by continuously re‑orthogonalizing the basis, converges in 30–40 iterations to a solution with residual norm below (10^{-6}).

In conclusion, the paper provides a rigorous and practically effective unifying framework for doubling algorithms. By eliminating the restrictive basis‑form assumption and introducing a dynamically updated orthogonal transformation, the Q‑doubling algorithm achieves superior robustness and broader applicability. This advancement is poised to impact a wide range of fields—control theory, quantum mechanics, electromagnetic modeling, and any domain where nonlinear matrix equations or structured eigenvalue problems arise.