A complete pair of solvents of a quadratic matrix pencil

TL;DR

The paper investigates the concept of complete pairs of solvents for quadratic matrix pencils, which facilitate solving second-order differential equations efficiently by reducing them to matrix exponential calculations.

Contribution

It introduces the notion of complete pairs of solvents for quadratic matrix pencils and discusses how to find such pairs to minimize numerical errors in differential equation solutions.

Findings

Complete pairs enable reduction to matrix exponentials.

Method for finding solvents with minimal rounding errors.

Improved stability in solving second-order differential equations.

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.