A complete pair of solvents of a quadratic matrix pencil

TL;DR

This work studies the second-order matrix differential equation $x''(t)+Bx'(t)+Cx(t)=f(t)$ through the quadratic pencil $L(\lambda)=\lambda^2 I+\lambda B+C$ and right solvents $X$ satisfying $X^2+BX+C=0$. A complete pair $(X,Z)$, with invertible $X-Z$, enables reducing the IVP solution to two matrix exponentials via $U(t)=(e^{Xt}-e^{Zt})(X-Z)^{-1}$, $U'(t)=(Xe^{Xt}-Ze^{Zt})(X-Z)^{-1}$; this motivates a spectral algorithm for finding complete pairs and a systematic study of numerical conditioning. The authors show that complete pairs correspond to invariant subspaces of the companion matrix $\mathcal{C}_1$ and that $\psi(f)$ and $U(t),U'(t)$ can be expressed in terms of two linear pencils, significantly simplifying computation. Through extensive numerical experiments across Hermitian, skew-Hermitian, and random pencils, they demonstrate that the numerical quality of the solution hinges on the conditioning of the solvents and their difference, rather than solely on spectrum-splitting heuristics, and they provide practical guidance for selecting the most stable complete pair. This work advances practical methods for solving quadratic matrix pencils with improved numerical robustness in the computation of matrix exponentials.

Abstract

Let $B$ and $C$ be square complex matrices. The differential equation \begin{equation*} x''(t)+Bx'(t)+Cx(t)=f(t) \end{equation*} is considered. A solvent is a matrix solution $X$ of the equation $X^2+BX+C=\mathbf0$. A pair of solvents $X$ and $Z$ is called complete if the matrix $X-Z$ is invertible. Knowing a complete pair of solvents $X$ and $Z$ allows us to reduce the solution of the initial value problem to the calculation of two matrix exponentials $e^{Xt}$ and $e^{Zt}$. The problem of finding a complete pair $X$ and $Z$, which leads to small rounding errors in solving the differential equation, is discussed.

Figures (8)

• Figure 1: The best (left) and the worst (right) splittings of the spectrum of $\mathcal{C}_1$ from Example \ref{['ex:1']}(a). The points of the spectrum of $\mathcal{C}_1$ related to $X$ are depicted by points, and related to $Z$ are depicted by stars
• Figure 2: The best (left) and the worst (right) splittings of the spectrum of $\mathcal{C}_1$ from Example \ref{['ex:1']}(b). The points of the spectrum of $\mathcal{C}_1$ related to $X$ are depicted by points, and related to $Z$ are depicted by stars
• Figure 3: The best (left) and the worst (right) splittings of the spectrum of $\mathcal{C}_1$ from Example \ref{['ex:1']}(c). The points of the spectrum of $\mathcal{C}_1$ related to $X$ are depicted by points, and related to $Z$ are depicted by stars
• Figure 4: The best (left) and the worst (right) splittings of the spectrum of $\mathcal{C}_1$ from Example \ref{['ex:2']}(a). The points of the spectrum of $\mathcal{C}_1$ related to $X$ are depicted by points, and related to $Z$ are depicted by stars
• Figure 5: The best (left) and the worst (right) splittings of the spectrum of $\mathcal{C}_1$ from Example \ref{['ex:2']}(b). The points of the spectrum of $\mathcal{C}_1$ related to $X$ are depicted by points, and related to $Z$ are depicted by stars

Theorems & Definitions (51)

• Definition 1
• Proposition 1
• proof
• Theorem 2: see, e.g., Kurbatova-POMI:eng
• Definition 2
• Definition 3
• Proposition 3
• proof
• Theorem 4: Gantmakher59:eng, Krein-LangerI:eng
• Definition 4