# Numerical solutions to large-scale differential Lyapunov matrix equations

## TL;DR

This paper introduces two novel numerical methods for solving large-scale differential Lyapunov matrix equations with low rank terms, combining integral approximations and Krylov subspace projections to improve efficiency and accuracy.

## Contribution

The paper proposes two new approaches for solving large-scale differential Lyapunov equations, integrating integral approximations and Krylov subspace techniques.

## Key findings

- New theoretical results on differential Lyapunov equations
- Numerical experiments demonstrating method effectiveness
- Efficient algorithms for large-scale problems

## Abstract

In the present paper, we consider large-scale differential Lyapunov matrix equations having a low rank constant term. We present two new approaches for the numerical resolution of such differential matrix equations. The first approach is based on the integral expression of the exact solution and an approximation method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low-dimensional differential Lyapunov matrix equation. The latter differential matrix problem is then solved by the Backward Differentiation Formula method (BDF) and the obtained solution is used to build the low rank approximate solution of the original problem. The process being repeated until some prescribed accuracy is achieved. We give some new theoretical results and present some numerical experiments.